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Efficiency droop in indium gallium nitride light emitters: an introduction to photon quenching processes

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Efficiency droop in indium gallium nitride light emitters: an introduction to photon quenching processes

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Content EFFICIENCY DROOP IN INDIUM GALLIUM NITRIDE LIGHT EMITTERS: AN INTRODUCTION TO PHOTON QUENCHING PROCESSES by Raymond Sarkissian A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2013 Copyright 2013 Raymond Sarkissian Acknowledgments I thank my advisor, professor John O'Brien, for adopting me in his group. He provided the means to capture and nurture science, for which I am very grateful. I also thank professor Daniel Dapkus who gave me the opportunity to work on the eciency droop problem for InGaN light emitters and made it possible for that research to ourish and grow. I thank professor Stephen Bradforth for sharing his lab that enabled all the experimental work in this manuscript pertaining toInGaN-based light emitters. I thank Dr. Sean Roberts and Dr. Ting-Wei Yeh who closely collborated with me on the e- ciency droop research. I also thank all former and current students of professor O'Brien's group specially Dr. Stephen Farrell. I also thank professor James Speck of UCSB and Dr. Kathryn kelchner for their collaboration in the eciency droop research. When I rst joined the Ph.D. program at Electrophysics department of school of engi- neering I had one wish: To sustain the excellence that many had brought in and fostered before me. Today, I feel I have contributed towards fullling that wish. ii Table of Contents Acknowledgments ii List of Tables iv List of Figures v Abstract viii Chapter 1: Introduction 1 Chapter 2: The optical setups 12 Chapter 3: Photon quenching in InGaN light emitting devices 18 Chapter 4: Photon quencing in an m-plane InGaN light emitter. 29 Chapter 5: Dynamical Stark eect in an m-plane InGaN light emitter 37 Chapter 6: Laser dynamics: Probing microscopic processes inInGaN light emitters. 46 Chapter 7: Photonic crystals: A short introduction 63 Chapter 8: An ecient four-port photonic crystal ber taper coupler 67 Chapter 9: Group index oscillations in photonic crystal waveguide resonators 81 Chapter 10: Nonliear switching of optical resonances 85 Chapter 11: Device fabrication 94 11.1 Silicon device fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 11.2 InGaN/GaN device fabrication . . . . . . . . . . . . . . . . . . . . . . . . 96 Bibliography 98 iii List of Tables 6.1 Denition of the variables in the semiconductor-Bloch equations . . . . . 62 iv List of Figures 1.1 Output power, p 0 , and eciency, ex , for a high eciency LED and a conventional LED reported in [NNS + 06] . . . . . . . . . . . . . . . . . . . 3 1.2 Warm-white pc-LED package loss channels and eciencies [DOE13] . . . 4 1.3 Warm-white cm-LED package loss channels and eciencies [DOE13] . . . 5 1.4 Task A.1.2 addresses the need for an improved understanding of the criti- cal materials issues impacting the development of higher-eciency LEDs. A key focus will be on identifying the fundamental physical mechanisms underlying the phenomenon of current droop in high-performance blue LEDs. Another focus will be on improving IQE and reducing the thermal sensitivity of LEDs, especially those in the red and amber spectral regions [DOE13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Demonstration of good ts obtained by both carrier escape and Auger recombination models for a) Nichia LEDs and b)Lumileds LEDs [ OLL + 10, SXD + 09] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Normalized photon ux spectra of structures 1,2 and 3 for a) 1:82A=cm 2 and b) 90:9A=cm 2 [VIK + 09]. . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 The measured photo-excited external quantum eciency, absolute inter- nal quantum eciency, and carrier density of sample A (open diamonds, thin solid line, and thick solid line) and sample E (closed circles, dashed thin line, and dashed thick line vs excitation level. In the inset, a Poisson- Schrodinger simulated band digram shows that at band conditions pre- vail. Ref. [SMW + 07] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 The transient absorption measurement setup . . . . . . . . . . . . . . . . 13 2.2 Time correlated single photon counting setup . . . . . . . . . . . . . . . . 15 2.3 The tapered ber photonic crystal waveguide coupler optical setup . . . . 16 2.4 The tapered ber photonic crystal waveguide coupler in the setup . . . . 17 3.1 mOD spectra for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . . . . . . . . . . . . . . . . . . . 20 3.2 mOD Temporal behavior ofmOD for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . . . . . . . 21 3.3 Transient photoluminescence of the InGaN quasi quantum well sample at an estimated carrier density of 2:34 10 18 cm 3 . The inset depicts the same transient photoluminescence for 500 ps. . . . . . . . . . . . . . . . . 22 3.4 mOD Temporal behavior ofmOD for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . . . . . . . 24 v 3.5 Quantum eciency, Np Nc , of the c-plane InGaN quasi-quantum well sample. It is evident that the eciency at 420nm drops as the carrier density levels rise in this structure showing a droop-like behavior. . . . . . . . . . . . . . 27 4.1 DierentialmOD for the single quantum well m-plane InGaN sample at an estimated carrier density of 4:7 10 19 cm 3 . The pump pulse energy was 120nj. The oscilltory feature at 390nm is due to ac Stark eect. . . . 30 4.2 Transient photoluminescence of the m-plane InGaN single quantum well sample excited by 400nm pulses. The estimated carrier density is 5:68 10 19 cm 3 for a pulse energy of 145nj. . . . . . . . . . . . . . . . . . . . . 31 4.3 Quantum eciency, Np Nc , of the m-plane InGaN single quantum well sam- ple. It is evident that the eciency at 420nm roles over at a carrier density between 1:6 10 19 cm 3 and 5:6 10 19 cm 3 . . . . . . . . . . . . . . . . 34 4.4 Transient photoluminescence of the m-plane InGaN single quantum well sample under a higher excitation density corresponding to an estimated carrier density of 3:525 10 20 cm 3 . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Transient photoluminescence of the m-plane bulk GaN sample recorded at two wavelengths of 420nm and 500nm corresponding to the pump induced absorption and absorption bleaching signals in the dierential mOD data. The TCSPC signal at 500nm was much weaker and was recorded under 440nj excitation pulses. The TCSPC signal at 420nm was recorded under an excitation pulse energy of 70nj six times weaker than that of the 500nm signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.1 Static photoluminescence of the m-planeInGaN single quantum well and BulkGaN samples. The vertical axis corresponds toLog P sample Psource where P sample andP source are the photoluminescence of the sample and excitation powers respectively. With this prescription each unit on the vertical axis corresponds to one order of magnitude dierence in the emission intensity 40 5.2 Dierential mOD for the single quantum well m-plane InGaN sample at an estimated carrier density of 4:710 19 cm 3 . The pump pulse energy was 120nj. The oscilltory feature at 390nm is due to ac Stark eect. . . . 42 5.3 Dierential mOD for the single quantum well m-plane InGaN sample illustrating the evolution of ac Stark eect. Note how this eect starts as multiple spectral oscillations and lasts only a short while. . . . . . . . . . 43 5.4 Temporal dynamics of the dierentialmOD for the single quantum well m-plane InGaN sample probed for delay times as long as 950ps. . . . . . 43 6.1 Cross section views of the Ey (perpendicular to growth direction) compo- nent of the electric eld. Three modes are shown for a 10m microdisk containing 6 pairs of InGaN/GaN of quantum wells . . . . . . . . . . . . . 56 6.2 Calculated cavity Q for the three modes of the microdisk shown in Fig.6.1 57 6.3 Cross section views of the Ey (perpendicular to growth direction) com- ponent of the electric eld. The membrane is 240nm thick and the gap between the membrane and the substrate is 320nm . . . . . . . . . . . . . 58 6.4 Calculated optical spectrum of a GaN membrane L7 cavity in the prox- imity of a substrate with a gap of 160nm . . . . . . . . . . . . . . . . . . 58 vi 6.5 Calculated cavity Q for the mode of the L7 cavity at the center wavelength of 475nm versus the distance of the membrane and the substrate. . . . . 59 6.6 Calculated modulation response of a VCSEL [CSK + 02]. . . . . . . . . . . 60 6.7 Modulation response of a Nitride-based laser emitting at 450nm [MSS + 10]. 60 7.1 Please double click on the image to play. This movie was produced by performing an FDTD calculation on USC's High-Performance Computing and Communications (HPCC) super computer. . . . . . . . . . . . . . . . 64 8.1 An illustration of the technique used in fabrication of the photonic coupler under study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8.2 Top microscope view of tapered ber partially overlapping the waveguide defect region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.3 SEM image of a fabricated photonic crystal waveguide with lithographi- cally dened facets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.4 An illustration of the coupler. The black lines represent the ber and the red lines represent the waveguide . . . . . . . . . . . . . . . . . . . . . . . 70 8.5 Transmission spectrum of the unloaded waveguide and the t obtained via expansion by Lorentzians . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.6 FWHM and group index extracted by tting the spectrum of the unloaded waveguide resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.7 Measured output power from right waveguide facet when the input light is launched through the left facet. . . . . . . . . . . . . . . . . . . . . . . . 78 8.8 Measured output power from the right waveguide facet when the input light is launched via port 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.9 Coupling eciency and the measured output power from port 2 when the input light is launched through the left facet . . . . . . . . . . . . . . . . . 79 9.1 The experimentally extracted and average group indices . . . . . . . . . . 83 9.2 Extracted phase contributions of a photonic crystal waveguide resonator . 84 10.1 Probe transmission for dierent estimated coupled pump energy at 1600 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 11.1 Electron beam pattern used to expose the PMMA lm. . . . . . . . . . . 95 11.2 Eects of O2 ash procedure is demonstrated here. As can be seen sig- nicant improvement in reduction of unresolved PMMA residue can be achieved after developing without much eect on the overall thickness of the PMMA mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 11.3 An SEM image of patterned holes into a silicon membrane suspended in air: part of a fabricated photonic crystal waveguide resonator. . . . . . . . 96 11.4 SEM images of GaN nanowires before (on the left) and after (on the right) HSQ spin-deposition (images provided by Dr. Ting-Wei Yeh). . . . . . . 96 11.5 SEM images of nanowires aftre ICP etch using RayGaN recipe (on the left) and after removal of HSQ (on the right) HSQ. (images provided by Dr. Ting-Wei Yeh). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 vii Abstract This thesis contains work from two separate projects, a study of the eciency of light emitting diodes, and a tapered-ber approach to photonic crystal integrated photon- ics. The rst part of this thesis describes an experimental investigation of the quantum eciency of InGaN-based light emitters. Blue and Green LEDs that utilize InGaN quantum wells for their active medium suer from a reduction in eciency with increas- ing bias. This phenomenon is called eciency droop. In this thesis experimental evidence for signicant quenching of photon population in InGaN is presented and its relevance to the eciency droop problem in InGaN-based light emitting structures is discussed. An equilibrium rate equation model is set up to demonstrate that radiative eciency for this loss mechanism not only has a similar dependence on carrier density as Auger recombination process, but it also possesses the right order of magnitude making it dicult to distinguish between the two and possi- bly leading to errors in interpretation. The impact of photon quenching processes on device performance is emphasized by demonstrating loss of eciency for spectral regions where there is experimental evidence for photon quenching. We have observed this phe- nomenon for both c-plane and m-plane light emitting structures. Both structures exhibit droop-like behavior for spectral regions where there is evidence for photon quenching. We have also observed and characterized the dynamical Stark eect for an m-plane light emitter considered in this manuscript. Our results revealed localization centers with a corresponding band-edge energy of 388nm and an excitonic binding energy of 17.81mev. viii Furthermore, fabrication of a photonic crystal waveguide ber taper coupler is demon- strated with a peak coupling eciency of 97 %. All four ports of the device are accessible providing an opportunity for investigation of simultaneous interaction of dierent light sources inside the photonic crystal cavity. A numerical model is set forth to analyze such devices with an excellent agreement with the experimental data. One important result of that theory is the ability to experimentally extract the phase contribution of optical resonators that employ periodic structures such as photonic crystal cavities. This device has also been used to demonstrate all-optical nonlinear shift and bleaching of cavity reso- nances via non-degenerate two photon absorption, non-degenerate Kerr mechanism, free carrier absorption, and free carrier plasma eects. As the response time of two photon processes are very fast, about 10 fs, this device can be used in ultrafast low energy all optical switching applications. ix Chapter 1 Introduction This manuscript, in part, addresses some of the issues involved with considering semi- conductor based light emitters for general lighting. Aside from the technological and economical impact that such light sources may bring, from the mere energy conser- vation stand point, the future of lighting will be heavily impacted by semiconductor based light sources. It has been predicted that doubling or tripling the current con- version eciency of the light sources used for general lighting would reduce the energy consumption by more than 50-75 % reducing the generation capacity by 30-45 GW in the US alone [HT11]. By 1999 the capability of semiconductor-based light emitters in achieving eciencies reaching to 50 % was already recognized promising a bright future with a plethora of savings in energy. It is worth knowing that most light sources today are only 5-7 % ecient and consume 6.5% of the global energy produced [HT11]. The use of semiconductor-based light emitters for lighting is often referred to as solid state lighting (SSL). The choice of a proper semiconductor for general lighting, however, has not been an easy one to make [Nak95]. The red and yellow light emitting diodes (LED) were demonstrated rst using the AlGaInP and other material system. There was a need for commercially viable blue/green light emitting structure to make up the white light. Material systems based on II-VI compound semiconductors, possessing a wide band gap energy, were good candidates. Availability of a lattice matched substrate, GaAs, made these material systems specially attractive. However, due to the short life- time of most of the II-VI material systems and indirect band-gap of SiC, III-V Nitride material systems were introduced and the rst blue LEDs were successfully manufac- tured. Side-by-side placement ofAlGaInP andInGaN based light emitters to generate white light has proven to be challenging due to dierences in operating voltages and 1 maintaining a temperature independent white light [Nar04, HT11]. One solution, cir- cumventing this problem, is to use blue InGaN based LEDs to excite phosphor. This technology is currently heavily in use and is expected to be the dominant technology for the near future. Commercially available LED lamps manufactured by Phillips Inc. and Cree Inc. incorporate blue LEDs along side with phosphor to generate white light. There is a similar law to Moor's law when it comes to solid state lighting. That law is known as Haitz' law [HT11], which predicts that $ lm would drop by a rate of $10 lm =decade. It also predicts an increase in ux of LEDs by more than 20lm lamp =decade. Now the total conversion eciency can not be more than 100 %, which corresponds to 400 lm W [HT11]. On the other hand the total illumination needs of current human activity is set by the current lighting technology and is about 3lm-3000lm for residential and 1klm-50klm for commercial applications. This sets an upper bound on the lm lamp necessary, where as the eciency of LEDs should continue to rise and is predicted to reach 150 lm W -200 lm W by 2020. One major problem hindering the incorporation of InGaN-based LEDs into general lighting is eciency loss as the supplied electrical power is increased. This can be seen in Fig. 1.1 Eciency droop has one major consequence and that is an increase in energy consump- tion. As will be described later the cause of eciency droop is unknown. Because the cause of this problem is unknown, a major overhaul of the current technology may be needed before this problem can eectively be addressed. It is possible that the lm W may diverge from Haitz' law and saturate, or even increase, for some period of time in which the current solid state lighting technology platform undergoes this reform. Fig. 1.2 shows the break down of eciencies for phosphor converted LEDs (pc-LEDs) [DOE13]. Elim- inating the eciency droop problem will address a major portion of the nal eciency goal set by U.S. department of energy (DOE). Note that for this gure the following were assumed: 1. LED package eciencies are reported at 25 C and 35 A/cm. 2. The analysis assumes a CCT of 3000 K and CRI of 85. Dierent choices of CCT/CRI will lead to slightly dierent results. 2 Figure 1.1: Output power, p 0 , and eciency, ex , for a high eciency LED and a conventional LED reported in [NNS + 06] 3. The phosphor conversion eciency is an estimate over the spectrum including the loss due to the Stokes shift (90 percent quantum yield times the ratio of the average pumped wavelength and the average wavelength emitted). The value here is typical of a blue LED pumping a yellow and red (for warm-white) phosphor system. Other phosphor formulations will give dierent results. 4. The current droop from the peak eciency to that at the nominal current density is shown here as an opportunity for improvement, since there is still as much as a 15 percent gain in eciency to be had by eliminating this loss at 35 A/cm2, and much more if the diode is operated at higher currents. As mentioned earlier color mixed LEDs (cm-LED) have not yet enjoyed commercializa- tion. However, this can change as the problems with producing a pc-LED based white light are resolved. Ref. [HT11] suggests that the need for elctronically controlled white light that would provide dierent hue may interest enough consumers to let this tech- nology be economically viable. Fig. 1.3 presents the break down of the eciencies for cm-LEDs. Although the eciencies of the blue LEDs have been improved signicantly during the past decade the eciency droop problem still remains on the list of tasks 3 Figure 1.2: Warm-white pc-LED package loss channels and eciencies [DOE13] for core LED technology research and development [DOE13]. Fig. 1.4 summarizes task A.1.2 of the multi-year program plan provided by DOE. That task addresses the e- ciency droop problem and focuses on identifying key physical mechanisms responsible for the eciency droop problem in all color LEDs. The goal is to eliminate the eciency droop as the supplied electrical currents is increased to 100 A=cm 2 from 35 A=cm 2 . As mentioned earlier the physics behind the eciency droop is unknown and there have been many suggestions for its origin [Pip10]. For example it has been suggested that the Auger recombination process could deplete the carriers from the band edges, as their densities rise in the quantum wells, making them unavailable to the spontaneous emission. As can be seen from the Fig. 1.5, very good ts to experimental data have been obtained by assuming that the Auger recombination is the only mechanism behind the loss of eciency. On the other hand, it has been shown that escape of the carriers into the barriers could also cause a droop behavior. Further, it has been shown that this escape of carriers can be facilitated by the distortion of the band edges induced by the strong piezoelectric eld inherent to InGaN/GaN based quantum wells grown on c-plane substrates. Very good 4 Figure 1.3: Warm-white cm-LED package loss channels and eciencies [DOE13] ts to experimental data also have been obtained by taking the carrier escape mecha- nisms to be the only process behind the loss of eciency in InGaN based light emitting structures [ OLL + 10, SXD + 09], seen in Fig. 1.5. Finally, the tilt of the band edges across the InGaN quantum wells could also cause a y-over of electrically injected carriers which may also play a part in loss of eciency [VIK + 09]. Ref. [VIK + 09] reports on a study in which the light emission from a p-doped InGaN/GaN quantum well embedded in the p-GaN layer of an otherwise standard LED is investi- gate. The Indium content of the extra quantum well is less than that of the original gain region and it also incorporates an electron blocking layer of its own. Comparison of the normalized emission from a standard LED (structure 1) and two other structures that contain an extra quantum well in their p-GaN layers (structures 2 and 3) veries the existence of electron over ow as can be seen in Fig. 1.6. The structures 2 and 3 are essentially identical except for the electron energy of the electron blocking layer of the main gain region, which is smaller for structure 3 compared to that of the structure 2 by fty percent. 5 Figure 1.4: Task A.1.2 addresses the need for an improved understanding of the critical materials issues impacting the development of higher-eciency LEDs. A key focus will be on identifying the fundamental physical mechanisms underlying the phenomenon of current droop in high-performance blue LEDs. Another focus will be on improving IQE and reducing the thermal sensitivity of LEDs, especially those in the red and amber spectral regions [DOE13]. A previous experimental study conducted on a quasi-single quantum well structure con- cluded that Auger recombination was the likely reason behind the observed eciency droop since the light emitting structure under study was expected to exhibit no tilt or distortion of the band edges across the emitting region [SMW + 07]. Fig.1.7 illustrates the measured external quantum eciency alongside with the calcu- lated internal quantum eciency of quasi-bulk (10nm to 77nm thick) InGaN layers 6 Figure 1.5: Demonstration of good ts obtained by both carrier escape and Auger recombination models for a) Nichia LEDs and b)Lumileds LEDs [ OLL + 10, SXD + 09] Figure 1.6: Normalized photon ux spectra of structures 1,2 and 3 for a) 1:82A=cm 2 and b) 90:9A=cm 2 [VIK + 09]. grown on c-plane GaN with an emission peak of 440nm [SMW + 07]. The Auger recom- bination rate coecient extracted in this experiment for the samples A and E shown in Fig.1.7 are 2 10 30 and 1:4 10 30 respectively. It is known that rst-principles calculations predict an Auger recombination coecient that, even when interaction with phonons is taken into account, is orders of magnitude smaller than that obtained by t- ting experimental data [PKH + 09, BGB10, DRVdW09]. Recently it has been suggested that an indirect Auger recombination mechanism, where the carrier-carrier scattering coupled with phonons could scatter the electrons into the second conduction band in InGaN quantum wells, has a magnitude large enough to explain the eciency droop 7 [KRDVdW11] Figure 1.7: The measured photo-excited external quantum eciency, absolute internal quantum eciency, and carrier density of sample A (open diamonds, thin solid line, and thick solid line) and sample E (closed circles, dashed thin line, and dashed thick line vs excitation level. In the inset, a Poisson-Schrodinger simulated band digram shows that at band conditions prevail. Ref. [SMW + 07] Chapter 3 of this manuscript reports an experimental investigation in which the car- riers are optically excited within an InGaN quasi-quantum well. Because the carriers are optically excited within the quantum well, we did not expect to observe the carrier escape eects described in the previous paragraph. In that chapter observation of a signicant photon quenching for an InGaN light emitter is reported, the magnitude of which is large enough to have a signicant eect on LED device performance. After discussing the experimental results, an equilibrium rate equation model is pro- posed for this quenching and analyzed. One important result of this model is that under typical operating conditions, the photon quenching processes observed here have a cubic dependence on carrier density as do Auger processes. Therefore, relying on denitions of radiative eciency that are based on equilibrium rate equations may not be sucient to distinguish between the contributions of Auger recombination and photon quenching processes. In order to distinguish between processes that actually quench the photon population 8 and processes that redistribute the carriers, we have chosen to employ an ultrafast pump- probe spectroscopy of the samples in hand. Such an experimental technique provides a direct measurement of the probing light intensity before and after an optical excita- tion. Short-lived optical pulses and the ability to control the time between the excitation (pumping) and examination (probing) of the sample makes it possible to study the tem- poral behavior of microscopic processes. The experimental setup is discussed in chapter 2. Growing light emitting structures along the non-polar planes could eliminate the built- in piezoelectric eld and provide an opportunity for fabrication of high eciency LEDs. Understanding the dynamics of light emission, therefore, in m-plane InGaN based quan- tum well structures is important. Chapter 4 of this manuscript investigates the emission eciency for an m-plane InGaN single quantum well light emitter. In this structure no piezoelectric elds, ideally, are expected along the direction of connement. Instead of measuring the spectrally integrated eciency, as is normally done, we have chosen to study the eciency for certain spectral regions where both transient absorption and transient photoluminescence data suggest quenching of spontaneous population. As dis- cussed in chapter 3 photon quenching mechanisms do exist and with a magnitude large enough to signicantly impact performance of LEDs. It has been known that m-plane InGaN quantum well structures suer from compo- sitional inhomogeneities. This is particularly a major problem for non-polar InGaN quantum wells as higher composition of In is necessary to achieve the same peak emis- sion wavelength as their c-plane counterparts. [KCG + 11, LPM + 10] Fluctuations in In composition and well width provide a hospitable environment for a quasi-three dimen- sional connement. This complicates the emission dynamics not only by creating a route for localization of excitons but also by boosting excitonic absorption due to connement. Although the m-plane InGaN quantum wells may not have built-in piezoelectric elds in the direction parallel to the growth direction, the piezoelectric elds could exist in the plane of the quantum well due to bidirectional strain in that plane. On the other 9 hand free standing m-plane GaN substrates could have evolved facets in dierent crys- tallographic directions that not only contribute to In composition inhomogeneity but also could be a strong factor in generating strong piezoelectric elds that are present at the localization sites [KCG + 11, WPGD + 11]. Therefor experimentally identifying the exciton energies is very important. The dynamic Stark shift of the those energies is one way of gaining insight into material properties. It not only enables the experimentalist to investigate the microscopic processes but also helps supply valuable information about physical parameters that are of importance to device physicists and engineers. Chapter 5 of this manuscript reports upon observation of the dynamic Stark eect at 390nm for an m-plane InGaN quasi-quantum well light emitter. In chapter 6 microscopic rate equations are set up to study laser dynamics for InGaN- based light emitting structures. It is proposed that modulation response of such lasers may be used to experimentally extract microscopic scattering processes including Auger scattering process. The light extraction from LEDs is also under heavy investigation. This is particularly important knowing that photon quenching mechanisms, as discussed in chapter 3, impact the device performance at a microscopic level as well as macroscopic level. Ideally, a zero photon lifetime is preferred. Photonic crystals provide a platform to engineer devices where the active medium is conned to a small region and the light is only allowed to escape in a certain direction of the photonic lattice. High coupling eciencies have already reported for well-engineered cavity-waveguide interfaces [FWE + 07, LMO12]. On the other hand the light can be eciently coupled from a photonic crystal waveguide into an optical ber [SFO09] providing the opportunity for design and fabrication of novelInGaN based photonic crystal light emitters. Chapter 7 and Chapter 8 report on the fabrication and experimental investigation of a robust photonic crystal waveguide tapered ber coupler. Even though the coupler has been originally intended for optical integration, but the physics of the optical coupling between the cavity, waveguide, and the tapered ber remains the same from the mere optical standpoint. The photonic 10 crystal dimensions and the ber taper, of course, would need to be redesigned for GaN based devices. Chapter 7 gives a short introduction to photonic crystals and summerizes the chapters that follow. 11 Chapter 2 The optical setups The pump-probe spectroscopy setup incorporates a Ti-Sapphire laser that is frequency doubled using a BBO crystal to generate 150 fs pulses at 400 nm with a repetition rate of 1 kHz acting as the pump light. A part of the 800 nm light is delayed by a translation stage and used to produce a white light probe (320-950 nm) through super- continuum generation in a 2 m thick CaF 2 crystal plate [RMM + 12]. The CaF 2 plate is continuously rotated to to ensure the stability of the of the white light and also to prevent photo-damage to the crystal. The structure is then excited by the pump light and probed by the white light, which is detected at the output port of a spectrograph after transmission. The spectrograph disperses the probe onto a 256 pixel silicon array where it is then detected. A schematic of the this setup can be seen in Fig. 2.1. The translation stage provides delay times that can vary between 10 fs and 1ns. The resolution of the experiment, however, is limited by the temporal width of the pump and is around 150 fs. The pump light spot size is about 250 m in diameter. The probe is much smaller and is set to entirely t within the pump spot. The change in the optical density due to excitation is monitored by mOD =1000Log I WP I WOP whereI WP is the transmission intensity of the probe light when the pump is present and I WOP is the transmission of the probe when the pump is absent. With this prescription, positive values of mOD represent an increase in the absorption due to the presence of the pump and the negative values occur when there is a reduction of absorption or gain due to stimulated emission. The carrier densities within the quantum wells were estimated by comparing the I WOP values for the reference and target samples.That is ln I Target WOP I Reference WOP is a measure of the absorption coecient. However, it is important to note that this gives an upper bound 12 Figure 2.1: The transient absorption measurement setup for the carrier density as the re ections at the InGaN/GaN boundaries are ignored. This is becausejAj 2 = 1jRj 2 jTj 2 wherejAj 2 ,jRj 2 , andjTj 2 are the absorption, re ection and transmission of the probe through the InGaN layer. SettingjRj 2 to zero maximizesjAj 2 . One has to bear in mind that the re ection and absorption coecients are fundamentally related through causality. Therefore the above-mentioned method is only an approximation. A precise determination of the carrier densities would involve simultaneous measurements of the re ection and transmission of the probe light. Time resolved photoluminescence of the the samples have also been recorded as will be dis- cussed in later chapters. That setup incorporates a time-correlated single photon count- ing (TCSPC) system that employs a Ti-Sapphire laser operating at a 250kHz repetition rate. A parametric amplier (OPA) is used to center the excitation optical pulses at 400 nm. The spontaneous emission is recored at right a right angle to the sample after a 0.125 m double monochromator using a photomultiplier tube with a 20 ps instrument 13 response time. As the carrier lifetime of InGaN-based quntum well light emitters are expected to be a few hundred picoseconds or less, it becomes very dicult to directly sample the decay due to speed limitations of the electronics. For example, a few hundred picoseconds of decay time would require sampling times of the order of 50 ps, which is dicult to achieve with current electronics. TCSPC is a technique that allows for reconstruction of the emission decay prole by detection and time allocation of single photon emission events over many excitation-detection cycles. It is important to maintain a single photon detection scheme. This is because of the ability of the detector to temporally resolve con- secutive photon events. This means that the detector will not register cycles with more than one photon population leading to over-registration of early photons, a phenomenon known as photon pile-up in time resolved photoluminescence measurements using TCSPS technique. This is particularly important when the excitation density dependent mea- surements need to be carried out as is the case in chapter 4. To circumvent this problem we have used very well characterized neutral density lters to attenuate the light before the monochromator. The rule of thumb to make sure that most cycles will be single photon detection events is to make sure that the photon count rate at the detector is one to ve percent of the excitation laser's repetition rate. The setup operates under reverse start-stop mode, which means that the detection events are used to start a time to analog converter (TAC). The output of a photo diode that samples the excitation laser is then used to trigger o TAC, which constitutes the SYNC signal. Since the electrical pulsed generated by the PMT may vary in amplitude a constant fraction discriminator (CFD) is used to trigger TAC. Since the excitation laser's repetition rate is 250 kHz TAC must be able to measure limes as long as 4 s. This is too long for TAC/ADC to handle. To solve this problem the signal of the photo-diode or the SYNC signal corresponding to the excitation pulse is electronically delayed so that the signal from CFD could overtake the SYNC and trigger on TAC/ADC, which is then stopped shortly after by SYNC with a delay time that is previously known and set by the electronic delay. A schematic of 14 the setup can be seen in Fig. 2.2. The optical setup for photonic crystal waveguide ber taper measurements incorporates Figure 2.2: Time correlated single photon counting setup two tunable laser sources. One of these lasers is used to couple the light into the photonic crystal input facet after passing through an isolator, a collimator, and the input objective lens. The other tunable laser is used to launch the laser light into the photonic crystal waveguide using the ber taper. The output facet of the photonic crystal waveguide is probed by the output objective lens and spatially ltered, which is then recorded using electronically addressable detectors. The setup has the capability of recording all four outputs of the photonic crystal waveguide ber taper coupler discussed in chapter 8. All the measurements are automated and controlled via a LabView program written and maintained by the author of this manuscript. A block diagram of the setup is shown in Fig. 2.3 Fig. 2.4 shows a part of the optical setup that was used to conduct the experiment dis- cussed in chapter 8. The objective lens at the left is the input lens and is used to couple 15 Figure 2.3: The tapered ber photonic crystal waveguide coupler optical setup the light into the photonic crystal waveguide facet. The objective lens to the right is used to collect the light from the output facet of the waveguide. The optical ber, expect for the tapered region, runs perpendicular to the waveguide direction. The fabrication of the ber taper is performed by rst replacing the input lens with the torch. The tip of the torch is placed at about 5mm above the ber. The ber is pulled, from its two ends, using motorized stages while the torch performs oscillatory movements. The speed of oscillation is about 1cycle=s. The speed of oscillation, the distance of the torch from the ber and the speed of the motors stretching the ber are all adjusted to give an adiabatic taper. This is possible by constantly monitoring the optical transmission of the ber for best results. Adiabatic taper refers to kind of taper in which the diameter of the ber varies gradually along the taper from the maximum diameter to the minimum diameter. This gradual change in the geometry of the ber minimizes the re ections from the taper and allows for maximum transmission of power. After the ber taper fabrication is done the photonic crystal waveguide can be positioned below the ber taper where the nal alignment can be done. 16 Figure 2.4: The tapered ber photonic crystal waveguide coupler in the setup 17 Chapter 3 Photon quenching in InGaN light emitting devices This chapter provides experimental evidence for signicant quenching of the photon pop- ulation inInGaN light emitters and discusses its relevance to the eciency droop prob- lem inInGaN-based light emitting structures. An equilibrium rate equation model is set up to demonstrate that radiative eciency for this loss mechanism not only has a similar dependence on carrier density as Auger recombination process, but it also possesses the right order of magnitude making it dicult to distinguish between the two and possibly leading to errors in interpretation. The possible impact of the photon quenching pro- cesses on device performance is emphasized by demonstrating loss of eciency for spec- tral regions where there is experimental evidence for photon quenching. Since the advent ofInGaN quantum well based light emitting diodes (LED), the promise of LED powered general lighting and other applications that require high power optical throughput have been hindered by eciency droop. Eciency droop refers to a phenomenon in which the output power increases sub-linearly with current density, indicating a decrease in quan- tum eciency as carrier densities rise inside the quantum wells. The physics behind the eciency droop is much debated and there have been many suggestions for its origin [Pip10, OLL + 10, SXD + 09, VIK + 09, SMW + 07, Cho11]. Auger recombination processes have been suggested for the main cause of eciency droop and a signicant body of work on this matter can be found in the literature [SMW + 07, IMP + 13]. Escape of the carriers, facilitated by distortion of the band edges for c-plane structures, has been also suggested to counter the arguments for Auger recombination process [ OLL + 10, SXD + 09, VIK + 09]. 18 Many theoretical studies have been carried out to explain unusually large Auger recom- bination coecients, when compared to GaAs for instance, extracted from experimental data [PKH + 09, BGB10, DRVdW09, KRDVdW11]. Very recently, an experimental study [IMP + 13] recorded high energy electrons, by elec- tron emission spectroscopy, from an InGaN=GaN quantum well LED under electrical injection and suggested that Auger recombination was the main explanation for the observed phenomenon. As that observation correlated with previously recorded eciency data it was concluded that the Auger recombination is the main mechanism behind e- ciency droop. We have chosen to study a light emitting structure that is similar to that of [SMW + 07]. That experimental study conducted on a quasi-single quantum well structure concluded that Auger recombination was the likely reason behind the observed eciency droop since the light emitting structure under study was expected to exhibit no tilt or distor- tion of the band edges across the emitting region. This chapter reports an experimental investigation in which, like [SMW + 07], carriers are optically excited within an InGaN quasi-quantum well. Experimental evidence for signicant photon quenching is provided and shown that for spectrally specic regions where there is photon quenching there also is loss of eciency. It is argued that experimental data could be misinterpreted, when extracting an Auger recombination coecient, if the photon quenching processes are not taken into account. Also, it is argued that there are microscopic photon quenching processes that can be an alternative to Auger recombination and may have contributed to the observed phenomenon in [IMP + 13]. Our light emitting structure is composed of a 70 nmInGaN quasi quantum well grown on a c-plane 2m GaN buer layer that lies on top of a c-planeAl 2 O 3 substrate. A 100 nmGaN layer caps the structure. To distinguish the signal pertaining only to the quasi quantum well layer, we conducted the same studies on a bulk c-plane GaN grown on c-plane Al 2 O 3 substrate with a thickness matching that of the InGaN quasi quantum well structure. For future reference we refer to the quasi-quantum well structure as the 19 target sample and to the bulk GaN sample as the reference sample. XRD data reveals an In composition of 18 % for the target sample. In order to distinguish between processes that actually quench the photon population and processes that redistribute the carriers, we have chosen to employ an ultrafast tran- sient absorption spectroscopy of target and reference samples. Such an experimental technique provides a direct measurement of the probing light intensity before and after an optical excitation. Details of the transient absorption spectroscopy setup can be found in [RMM + 12]. The pump is tuned at 400nm with a temporal width of 150fs, a spot size of 250m in diameter, and directly excites the quasi quantum well. The probe is a white light with a spot size much smaller than that of pump. The resolution of the experiment is around 150 fs. The change in the optical density due to excitation is monitored by mOD =1000Log I WP I WOP [OKF + 03] where I WP is the transmission intensity of the probe light when the pump is present and I WOP is the transmission of the probe when the pump is absent. With this prescription positive values of mOD imply an increase in the absorption due to the presence of the pump and the negative values occur when there is a reduction of absorption or gain due to stimulated emission.The carrier densities within the quantum wells were estimated by comparing the I WOP values for the reference and target samples. Fig. 3.1 illustrates the evolution of mOD spectra 400 440 480 –12 –8 –4 0 4 Wavelength [nm] mΔOD 100 fs 1 ps 10 ps 100 ps 1 ns a) Figure 3.1: mOD spectra for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . 20 for an estimated carrier density of 10 18 cm 3 after accounting for Fresnel re ections at the air-material boundaries. Note how the signal at 415nm starts o as a negative one at early times, 100 fs, and evolves into a positive signal after about 335 ps, which can be seen in Fig. 3.2. The data reveals quenching of the probe photon population over a fairly broad spectral region, centered about 415 nm, with a magnitude comparable to regions where there is a negative change in the optical density due to excitation. The peak positive change in the optical density after 1ns reaches to 3.7, which translates into an excitation induced net absorption of about 529 cm 1 at an estimated carrier density of 1 10 18 cm 3 . We have investigated the transient photoluminescence of the target 0 400 800 1200 –12 –8 –4 0 4 delay time [ps] mΔOD 415 nm b) Figure 3.2: mOD Temporal behavior of mOD for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . and reference samples at variety of wavelengths including 420nm and veried that under similar excitation conditions there is spontaneous emission from spectral regions where the transient absorption data suggests photon quenching. This means that it is possible to absorb and annihilate a spontaneously emitted photon. Fig. 3.3 illustrates the transient photoluminescence of the target sample measured by a time correlated single photon counting (TCSPC) setup [RMM + 12]. The pump, like the transient absorption setup, is tuned at 400 nm with a temporal width of 150 fs. The time resolution of the system is about 20 ps as limited by the photomultiplier response. 21 It is interesting to use quasi-equilibrium rate equations to derive a radiative eciency Figure 3.3: Transient photoluminescence of the InGaN quasi quantum well sample at an estimated carrier density of 2:34 10 18 cm 3 . The inset depicts the same transient photoluminescence for 500 ps. where the photon quenching processes are taken into account as loss mechanisms. Let us imagine that spontaneous photons can be reabsorbed within a photon lifetime of the optical structure. In this model we treat the light emitting structure as a high index material surrounded by air. We can now assign an index, i, to each optical mode. Assum- ing an absorption identied bya, ignoring the Auger recombination process, we proceed by setting up the following: dN dt = X i i g a i N i p ANBN 2 +G (3.1) dN i p dt = i sp BN 2 N i p i p (3.2) Where i a , a i , N i p , A, B, N, G, i sp , and i p are the group velocity, the absorption coef- cient associated with an optical eigenmode of the index "i", the number of photons in that mode, the non-radiative recombination coecient, the radiative recombination 22 coecient, the carrier density, the generation rate of carriers, the portion of sponta- neous emission coupled to the optical mode "i", and the photon lifetime of that mode respectively. At the steady state condition N i p = i sp BN 2 i p (3.3) Taking the group velocity, absorption, and photon lifetime to be more or less the same for all the optical modes of the LED we can write the radiative eciency as r = BN 2 P i i sp AN +BN 2 + g a p BN 2 P i i sp (3.4) Now P i i sp = 1 so r = BN 2 AN +BN 2 + g a p BN 2 (3.5) The absorption, a, is carrier density dependent, which means that taking the rst term of the Taylor expansion of "a" would create a term in the denominator of the radia- tive eciency that has a cubic dependence on the carrier density. With the evidence presented here that photon quenching can be a signicant source for photon loss, this can be a source of error for extracting an Auger recombination rate coecient from the radiative eciency data since the latter also assumed to have a cubic dependence on carrier concentration. One important question to ask is the following: What constitutes photon quenching? In this letter all processes of the excited state of the light emitter that could quench the photon population either by absorption or inhibition of emission are characterized as photon quenching. One possible photon quenching process is via the direct dipole interaction of photons with electronic states of rst and second conduction bands in InGaN. It is interesting to note that for the material compositions where most of the droop data is concentrated, InGaN possesses a second energy gap between its rst and 23 second conduction bands that is equal to or smaller than its main gap between its rst conduction band and the valence band. This can be seen in Fig. 3.4, which is extracted from data presented in [DRVdW09]. High energy ellipsometry measurements, on the 400 500 600 –0.4 0.0 0.4 0.8 Eg [nm] Eg–Δ [ev] d) Figure 3.4: mOD Temporal behavior of mOD for the quasi quantum well c-plane InGaN sample at an estimated carrier density of 1 10 18 cm 3 . other hand, have shown that the second conduction band is mostly of p-type atomic orbital symmetry [RCE + 08]. This means that absorption of spontaneously emitted pho- tons from the rst conduction band can participate in transitioning electrons from the rst conduction band to the second conduction band. Such a mechanism can be one candidate describing the physics behind quenching of the photon population within the photon lifetime of the optical structure that we used to derive the radiative eciency above. We now estimate the magnitude of the third term in the denominator of Eq. 3.5 by assuming that the eective masses, joint densities of states, and the dipole matrix ele- ment are approximately equal for the valence band and the second conduction band, which means that the absorption associated with the rst to second conduction band transitions is equal to the gain associated with the stimulated emission from rst con- duction band to valence band transitions. Let's assume an absorption a = 2500cm 1 [MSS + 10] for a carrier density of 10 19 cm 3 , a quality factor (Q) of 100 at 450nm emission 24 wavelength, and a radiative recombination coecient B = 2 10 11 [SMW + 07]. With these parameters, and assuming a refractive index of 2.5 for the overall optical structure, we extracted an equivalent Auger recombination coecient (C) of 1:43 10 30 cm 6 s 1 assuming a functional form of CN 3 . On the other hand assuming a = 529cm 1 , which was extracted from the transient absorption data presented here, results an equivalent C coecient of 2:8 10 29 cm 6 s 1 . Both these numbers are within the range of C coef- cients reported in literature [SMW + 07, OLL + 10]. This stresses the fact that photon quenching mechanisms need to be taken into account when studying typical radiative eciency data. A spontaneous photon can also participate in the transition of the electrons from the rst conduction band to the second conduction band simultaneously. Statistics of such process would involve four occupation numbers and could be written asn1 c1 n2 c1 n3 c2 n4 v1 . Where n1 c1 and n2 c1 are, generally speaking, non-equilibrium distributions describing the statistics of the two electrons in the rst conduction band before the interaction. n3 c2 and n4 v , similarly, describe the statistics of the electrons in the second conduction band and the valence band respectively after the interaction. This microscopic process can be described by a two-particle Hamiltonian with two participating dipole-interaction terms. Since most of the states in the second conduction band are available one can set n3 c2 = 1. Therefore at equilibrium, and under similar excitation conditions as those reported for studies investigating the Auger processes, this process will also contribute as having a cubic carrier density dependence. The microscopic photon quenching pro- cess described here is an alternative to Auger recombination process and could have contributed to the results observed in [IMP + 13]. That study bases its conclusion on one major assumption that no other mechanism than Auger recombination could introduce high energy electrons recorded in the experiment. This assumption breaks down with the suggested mechanism outlined above. Also, higher energy electrons in the second conduction band could couple to continuum states and recombine further contributing to the eciency droop [Cho11]. 25 There may be other photon quenching processes. For example pump induced carriers could screen the piezoelectric elds present in a c-plane InGaN quantum well light emit- ter therefore changing the absorption spectrum of the excited state of the material by changing the oscillator strength of the emission. Also the excitonic absorption, that is sustained in room temperature for InGaN based light emitters could be amplied by such a mechanism resulting in a pump induced absorption in the mOD spectra [SDS + 00, OKF + 03, OKF + 02]. To our knowledge, there are no numerical models based on the microscopic picture of semiconductors that include all of the photon quenching processes mentioned above. However, the need for such numerical studies is apparent at this point. So far, experimental evidence for an ecient photon quenching has been provided. It is now important to study the loss of eciency. Instead of studying the radiative eciency by measuring the spectrally integrated emission we choose to study the loss of eciency at a specic point in the mOD spectra where both the transient absorption and tran- sient photoluminescence data suggest photon quenching. This is done, here, by measuring the transient photoluminescence of the target sample at 420nm under dierent excitation levels. We have chosen to evaluate the ratio of the total detected number of spontaneous photons and the total number of carriers, Np Nc . As can be seen from the Fig. 3.3 the transient photoluminescence signal goes through a maximum [LML + 10]. A rise time can be associated with this signal and the following equations can be set by assuming a generation rate G and an arbitrary carrier density dependent function called D (N c ) for carrier consumption rate. dN c dt = G D (N c ) (3.6) P sp = col sp BN 2 c ~!V (3.7) Where P sp , col , sp , B, N c , and V are the spontaneous emission power, the collection eciency, the percentage of the spontaneous emission that goes into the optical modes 26 being detected, the bimolecular recombination coecient, the carrier density, and the volume of decaying carriers. The internal quantum eciency is dened asIQE = BN 2 c D(Nc) . Taking the derivative ofP sp with respect to time, substituting for dNc dt from Eq.(6.1) and by setting dPsp dt to zero the following is achieved: N p N c = IQE col sp sp r (3.8) The total number of photons is the product of photon count rate and total integration time in the TCSPC measurement. The total number of carriers can be estimated by the knowledge of the carrier density and the pump spot size. sp and r are also readily extractable from the transient photoluminescence data. We have evaluated Np Nc for sev- eral dierent excitation levels and observed that for 420nm, where both the transient absorption and transient photoluminescence data indicate photon quenching, there is a signicant loss of eciency. As can be seen from the Fig. 3.5 the normalized value for Np Nc drops by 30 percent as the carrier density levels rise. 1.0 . 10 19 2.0 . 10 19 3.0 . 10 19 0.7 0.8 0.9 1.0 Carrier density [cm-3] Normalized Np/Nc Figure 3.5: Quantum eciency, Np Nc , of the c-plane InGaN quasi-quantum well sample. It is evident that the eciency at 420nm drops as the carrier density levels rise in this structure showing a droop-like behavior. In conclusion, we have observed experimental evidence for signicant quenching of the 27 spontaneous photon population in anInGaN light emitter over a broad spectral region. Transient absorption and transient photoluminescence data recorded under similar exci- tation conditions suggest that spontaneous photons can be eectively annihilated due to photon quenching processes and that the magnitude of such processes are such that they are expected to impact device performance. This is evident because the magnitude of the positive changes in the optical density of the excited state of the light emitting structure is comparable to the magnitude of the negative changes. Further investigation of the quantum eciency for those regions where both transient absorption and transient photoluminescence data suggest photon quenching, revealed a decreases in quantum e- ciency by about thirty percent. Quasi equilibrium rate equations were set up to demonstrate that not only such photon quenching processes have the same carrier density dependence as the Auger processes they also have the right order of magnitude to have resulted in miss-interpretation of typical experimental data for radiative eciency. Furthermore, we propose that transi- tioning of the electrons from the rst conduction band to the second conduction band by reabsorption of spontaneous photons, either simultaneously or within the photon life- time of the structure, can be a possible photon quenching process contributing to the loss of eciency both at microscopic and macroscopic levels. Detailed analysis of the transient absorption (mOD) spectra and the dynamics of evolution of those spectra require a theory based on full microscopic picture of InGaN light emitting structures. That theory may include a two particle Hamiltonian describing the interaction of the spontaneous photons with the two conduction bands of InGaN, Auger recombination, and piezoelectric eld eects all under the same numerical footing. In order to distinguish between the contributions of the dierent possible photon quench- ing mechanisms, further investigation is necessary. 28 Chapter 4 Photon quencing in an m-plane InGaN light emitter. This chapter report on the observation of photon quenching for an m-plane InGaN quantum well emitter. Growing light emitting structures along the non-polar planes could eliminate the built- in piezoelectric eld and provide an opportunity for fabrication of high eciency LEDs. Understanding the dynamics of light emission in m-plane InGaN based quantum well structures is therefore important. It has been known that m-plane InGaN quan- tum well structures suer from compositional inhomogeneities. This is particularly a major problem for non-polar InGaN quantum wells as higher composition of In is necessary to achieve the same peak emission wavelength as their c-plane counterparts. [KCG + 11, LPM + 10] Fluctuations in In composition and well width provide a hospitable environment for a quasi three dimensional connement. This complicates the emission dynamics not only by creating a route for localization of excitons but also by boosting excitonic absorption due to connement. Although the m-plane InGaN quantum wells may not have built-in piezoelectric elds in the direction parallel to growth direction the piezoelectric elds could exist in the plane of quantum wells due to bidirectional strain in that plane. On the other hand free standing m-plane GaN substrates could have evolved facets in dierent crystallographic directions that not only contribute to In composition inhomogeneity but also could be a strong factor in generating strong piezoelectric elds that are present at the localization sites [KCG + 11, WPGD + 11]. This chapter investi- gates the emission eciency for an m-plane InGaN single quantum well light emitter 29 where, ideally, no piezoelectric elds are expected along the direction of connement. Instead of measuring the spectrally integrated eciency, as is normally done, we have chosen to study the eciency for certain spectral regions where both transient absorption and transient photoluminescence data suggest quenching of spontaneous population. As discussed in chapter 3, photon quenching mechanisms do exist and with a magnitude large enough to signicantly impact performance of LEDs. These processes could impact the eciency at microscopic as well as macroscopic levels. Fig. 4.1 illustrates the dierential mOD spectra for a time delay of 5ps. The early stages of the spectrum exhibited evidence for the dynamic stark shift and bleaching of the an excitonic resonance and it is discussed elsewhere in chapter 5. The transient absorption data, as seen in Fig.4.1, reveals a pump induced absorption that peaks around 420nm. For those spectral regions where the transient absorption data indicates pho- ton quenching, we have also investigated the transient photoluminescence of the sample using a time correlated single photon counting (TCSPC) setup. The TCSPC technique, Figure 4.1: Dierential mOD for the single quantum well m-plane InGaN sample at an estimated carrier density of 4:7 10 19 cm 3 . The pump pulse energy was 120nj. The oscilltory feature at 390nm is due to ac Stark eect. besides enabling us to further study the dynamics of emission from our samples, provides a laboratory setting that is compatible with the pump-probe spectroscopy setup. This is 30 because the TCSPC setup also incorporates a frequency doubled Ti-Sapphire pulsed laser that produces a train of pulses at 400nm with a repetition rate of 250KHz and a pulse width of 150fs. The spontaneous emission is detected by a photomultiplier (PMT) after a monochromator. The time resolution of the system is about 20ps. In order to perform an excitation density dependent measurement while preventing the photon pile-up at the detector we utilized well characterized neutral density lters after the collection lens. Fig. 4.2 depicts the transient photoluminescence of the target sample at 420nm and Figure 4.2: Transient photoluminescence of the m-plane InGaN single quantum well sample excited by 400nm pulses. The estimated carrier density is 5:68 10 19 cm 3 for a pulse energy of 145nj. under a similar excitation conditions as that of Fig. 4.1. Therefore, the spontaneously emitted photons at 420nm can be absorbed and annihilated. As described in detail in chapter 3 the quenching of the photon population could signicantly impact the device performance and may be a major contributor to the loss of eciency for InGaN based light emitters. It is, therefore, important to investigate the eciency for those spec- tral regions where both the transient absorption and transient photoluminescence data suggest photon quenching. We have chosen to evaluate the ratio of the total detected number of spontaneous photons and the total number of carriers, Np Nc . As can be seen from the Fig.4.2 the transient photoluminescence signal goes through a maximum and 31 a rise time can be associated to the signal. We use this empirical fact and set the fol- lowing equations. Let us assume that the carriers are generated with a certain rate G. The rate G does not refer to the carriers generated right after the pump pulse rather it corresponds to the mechanism behind the rise time in the transient photoluminescence. The decay of the carriers can be described by an arbitrary carrier density dependent function called D (N). One could then write the following: dN dt = G D (N) (4.1) P sp = col sp BN 2 ~!V (4.2) Where P sp , col , sp , B, N, and V are the spontaneous emission power, the collection eciency, the percentage of the spontaneous emission that goes into the optical modes being detected, the bimolecular recombination coecient, the carrier density, and the volume of decaying excitons. The internal quantum eciency is dened asIQE = BN 2 D(N) . Taking the derivative of P sp with respect to time and substituting for dN dt from Eq.(6.1) the following is achieved: dp sp dt = 2 col sp BN (GD (N)) ~!V (4.3) dP sp dt = 2 col sp BN 2 N ~!V G BN 2 IQE (4.4) 1 P sp dP sp dt = 2 N G P sp col sp~!V IQE (4.5) 32 Eq.(6.4) and Eq.(6.5) are merely rearrangements of Eq.(6.3). Setting dPsp dt to zero forces the term in the parenthesis in Eq.(6.5) to be zero and hence: IQE col sp = P peak sp ~! GV (4.6) IQE col sp = Np sp Nc r (4.7) Where P peak sp , N p , sp , N c , r are the spontaneous emission power when the transient photoluminescence goes through the maximum, the total number of detected sponta- neous photons, the spontaneous emission life-time, the total number of carriers, and the rise-time of the transient photoluminescence signal. Then it is easy to see that: N p N c = IQE col sp sp r (4.8) The total number of photons is the product of photon count rate and total integration time in the TCSPC measurement. The total number of carriers can be estimated by the knowledge of the carrier density and the pump spot size. sp and r are also readily extractable from the transient photoluminescence data. We have evaluated Np Nc for sev- eral dierent excitation levels and observed that for 420nm, where both the transient absorption and transient photoluminescence data indicate photon quenching, there is a signicant loss of eciency. As can bee seen from the Fig. 4.3 the Np Nc drops to almost the quarter of its peak value as the carrier density levels are increased. In conclusion, we investigated the eciency of an m-plane InGaN single quantum well light emitter and observed a signicant loss of eciency for a spectral region for which there is a strong evidence for quenching of the spontaneous photon population. This study was motivated based on a previous letter that suggests the quenching of 33 Figure 4.3: Quantum eciency, Np Nc , of the m-plane InGaN single quantum well sample. It is evident that the eciency at 420nm roles over at a carrier density between 1:6 10 19 cm 3 and 5:6 10 19 cm 3 the spontaneous photon population may be a signicant contributer to the loss of e- ciency. We now brie y describe some of characteristics of the photoluminescence data observed here. The clear rise time of the transient photoluminescence has been observed before and attributed to diusion of the carriers into localization centers [LML + 10]. At higher excitation densities a longer decay mechanism seems to take eect as is evi- dent from Fig.4.4 where the carrier densities were estimated to be 2:73 10 19 cm 3 and 3:52510 20 cm 3 . This would be consistent with the carrier localization picture as band lling of lower density of states corresponding to localized centers would lead to spill-over of carriers to neighboring regions where the emission is less ecient primarily due to loss of connement. Fig.4.5 shows the transient photoluminescence of the reference sample at 420nm and 500nm. The emission at 420nm has a similar decay dynamics as that of the 500nm although it was recorded at a much lower excitation density. A simple t of the TSCPC data for the reference sample at 420nm gives a decay time of 13.56ns. Fix- ing this lifetime and tting the transient photoluminescence of InGaN single quantum well with a double exponential resulted into a decay time of 534ps, which we believe is associated with carrier conning structures. This further forties the idea that the longer tail in the target sample's photoluminescence comes from an emission mechanism 34 Figure 4.4: Transient photoluminescence of the m-plane InGaN single quantum well sample under a higher excitation density corresponding to an estimated carrier density of 3:525 10 20 cm 3 that is most likely common between the InGaN quantum well and the bulk GaN. This certainly would be the case if the connement of carriers were mostly possible through 3D quantum dot type structures rather than the 2D quantum well layer with potential variations. 35 Figure 4.5: Transient photoluminescence of the m-plane bulk GaN sample recorded at two wavelengths of 420nm and 500nm corresponding to the pump induced absorption and absorption bleaching signals in the dierential mOD data. The TCSPC signal at 500nm was much weaker and was recorded under 440nj excitation pulses. The TCSPC signal at 420nm was recorded under an excitation pulse energy of 70nj six times weaker than that of the 500nm signal. 36 Chapter 5 Dynamical Stark eect in an m-plane InGaN light emitter This cahpter reports upon observation of the dynamic Stark eect at 390nm for an m- plane InGaN quasi-quantum well light emitter. We apply ultrafast transient absorption spectroscopy techniques to investigate the carrier and emission dynamics from a 10nm InGaN single quantum well grown on an m-plane freestanding 330m GaN substrate. The light emitting structure also includes a 2m unintentionally doped GaN buer layer and an unintentionally doped 100nm capping layer. For future referencing we will call this the target sample. We also investigated a reference sample that is just an m-plane bulk GaN with a matching thickness as that of the target sample but with the quasi-quantum well and capping layers removed. The transient absorption spectroscopy setup incorporates a Ti-Sapphire laser that is frequency doubled through a BBO plate to generate 150fs pulses at 400nm with a rep- etition rate of 1KHz acting as the pump light. A part of the 800nm light is delayed by a translation stage and used to produce a white probe light through super-continuum generation in a CaF 2 crystal plate. The structure is then excited by the pump light, probed by the white light, and then, after transmission, detected at the output port of a monochromator. The pump is spatially ltered to prevent the pump scatter in the detector. The translation stage provides delay times that can vary between 10fs and 1ns. The resolution of the experiment is about 150fs. The pump light spot size is about 250m. 37 In order to account for nonlinear propagation eects through the relatively thick sub- strate due to two photon absorption we have also evaluated themOD for the reference sample under identical conditions to that of the target sample. ThemOD of the target and reference samples are then directly subtracted. We will refer to this value as the "dierential mOD". The dynamic Stark eect, also called ac-Stark eect, is a short lived phenomenon lasting about a few hundred femtoseconds. Both experimental and theoretical studies inves- tigating this phenomenon can be found in the literature [KPL88, SRCH88, CKM + 89, CLG + 02]. [KPL88] outlines a theory of transient nonlinearities assuming that collisions are negligible under non-resonant excitation conditions. In that study the dierential transmission signal, given by the normalized dierence of the probe tranmission for pump-ON and pump-OFF conditions, is investigated. For a small change in transmis- sion of the probe this signal measures the interference between the unperturbed and perturbed probe light. The dierential transmission signal is expected to show oscilla- tory features around the exciton resonance for negative delay times with a period equal to 1 tp wheret p is the delay time between the pump and the probe. These oscillations are expected to disappear leaving a dispersive shape due to the blue shift and bleaching of the exciton absorption spectrum as the delay time approaches zero. Under the above- mentioned condition the mOD signal is proportional to the dierential transmission and we expect to see similar results. The dynamical Stark eect is a non-equilibrium phenomenon and the non-equilibrium Green's function theories are suitable for numerical investigations. One such study is outlined in [SRCH88]. In that study it is shown that the dynamic Stark eect induced by the excitations below the exciton resonance can be understood by interaction of the probe with virtual excitons. It is shown that not only the exciton resonance may be renormalized by the pump, but the band edge energies also could experience a shift. A blue shift of the band edge energies would eectively shift the absorption edge towards higher energies. This will appear as a bleaching of the absorption and would show up 38 as a negative shift in mOD signal. It is also important to recall that InGaN is a piezoelectric material and the piezoelectric elds present on the sites of the excitons at localization centers may play a signicant role in this dynamics. As reported in [CKM + 89] an applied DC electric eld were observed to decrease the ac-Stark eect. The exciton absorption spectrum is expected to redshift and experience bleaching under piezoelectric elds. Ref. [CKM + 89] also reported oscillatory features in the dierential probe transmission for the negative delay times that evolved into a dispersive shape later on. Fig.5.1 shows the normalized static photoluminescence of the target and reference samples excited via a broad band incoherent source (Xenon lamp). Two peaks are clearly visible. A similar emission peak to that of the target sample at 389nm has also been observed for c-planeInGaN samples and attributed to emission coming from areas near defects.[HKL + 05]The broad emission that peaks at 550nm is the so-called yellow band emission and is about two orders of magnitude stronger for the target sample compared to that of the reference sample. Note that the emission at 389nm also exists for the reference sample however it is two orders of magnitude weaker. In order to gain further insight into the emission dynamics we have studied the transient absorption of both the target and reference samples excited at 400nm. Fig. 5.2 illustrates the dierentialmOD spectra for two delay times of 100fs and 5ps. The estimated carrier density is 4:7 10 19 cm 3 . Majority of these carriers are expected to be at regions with a band gap energy less than that of the pump photon energy. The dispersive feature at 390nm, considering that the pump is at 400nm, we think is due to the ac-Stark Shift of an excitonic resonance. The early eects of the ac-Stark shift appear as oscillations in themOD spectra for negative delay times, which vanish within a few hundred femtoseconds and can be seen in Fig. 5.3. In the limit of zero 39 Figure 5.1: Static photoluminescence of the m-plane InGaN single quantum well and Bulk GaN samples. The vertical axis corresponds to Log P sample Psource where P sample and P source are the photoluminescence of the sample and excitation powers respectively. With this prescription each unit on the vertical axis corresponds to one order of magnitude dierence in the emission intensity exciton interaction, early stages of the pump-probe spectra, as described in [KPL88], can be understood by the following: T (!) / 2 4 2 2 + 2 Re [A] +Re [B] (5.1) Where A and B are given by: A = Z tp 1 dt Z t 1 dt0 (5.2) E L (t) E (t0)e [i( e L )(tt0)] 40 B = 1 i Z 1 0 dtE L (t +t p ) e [i(! L )] (5.3) Z t+tp 1 dt0E (t0)e [i( e L )(tpt0)] In these equationsT (!),, andE L are dierential transmission, dipole matrix element, and the pump electric eld. Whereas, , , e , L , and ! are exciton linewidth, probe detuning from the exciton resonance given by =! e , exciton resonance frequency, pump eld frequency, probe eld frequency. The term A is independent of ! and all the ! dependence is contained in the B term. Therefore the oscillations of the dieren- tial transmission are also contained in this term. The prefactor 1 i peaks at = 0, which ensures that the oscillations and the dispersive signal at later times are around the exciton resonance. As can be seen from Fig. 5.3 the early oscillations merge into a dispersive signal right about 390nm, which is where the exciton resonance is predicted to be by the aforementioned equations. These equations also predict that the period of spectral oscillations for negative delay times (t p < 0) values is just 1 tp . Two photon generated carriers generated by an intense pump can create a non- equilibrium three dimensional distribution that can screen and eectively bleach the excitonic absorption due to many body eects. The eect of such many body interac- tions is similar to the case where the pump is tuned, very high in energy above the band gap, which is also discussed in the theory of [CKM + 89]. That theory also predicts early oscillations of the dierential transmission signal around the exciton resonance that instead of merging into a dispersive signal merge into a symmetric signal with a central peak at the exciton resonance frequency. This, in turn, can distort the oscil- lations at negative delay times as well as positive, but short, delay times. Therefore in order to quantify the dynamic stark shift of the exciton resonance accurately, the two-photon-absorption mediated bleaching of the exciton resonance has to be accounted for. The data of Fig. 5.2, however, suggests that the dynamic stark eect observed 41 Figure 5.2: Dierential mOD for the single quantum well m-plane InGaN sample at an estimated carrier density of 4:7 10 19 cm 3 . The pump pulse energy was 120nj. The oscilltory feature at 390nm is due to ac Stark eect. here is dominated by the generation of virtual excitons as the oscillations turn into a dispersive signal and not a symmetric signal at this level of excitation. This situation can change for higher excitation levels. The non-equilibrium distribution of real carriers generated by two-photon absorption could also thermally relax down to lower energies and contribute to the absorption bleaching by stimulated emission. However such eects are expected to take eect at much longer time scales since the electron scattering due to interaction with LO-phonons are in the order of picoseconds. As can be seen from the Fig. 5.3, the dispersive signal associated with the ac-Stark shift of the excitonic resonance is superimposed on an evolving negative spectrum that spans from 370nm to about 410nm. This spectrally broad negative change in optical density that takes only 300fs to develop is most likely mediated by carrier-carrier scat- tering rather than slower carrier-phonon scattering mechanisms. The dispersive signal at 390nm disappears once the pump has vanished, but the aforementioned spectrally broad negative signal is long lived as can be seen from Fig.5.4. This is an evidence for existence of real carriers in the sample under study as the virtual carriers only exist for only a few hundred femtoseconds. It is also important to know that the pump could 42 Figure 5.3: Dierential mOD for the single quantum well m-plane InGaN sample illustrating the evolution of ac Stark eect. Note how this eect starts as multiple spectral oscillations and lasts only a short while. also be absorbed through an Urbach's tail below the band gap energy. This is possible because of large LO-phonon energies ranging between 72mev and 92mev corresponding to InN and GaN respectively.[KAV + 96, ZBR + 04] For example a pump photon at 400nm could scatter a virtual exciton into a real exciton at an energy corresponding to 388nm by interaction with a phonon of 92mev in energy. The dynamic Stark shift of the exciton resonance can be described by the following. Figure 5.4: Temporal dynamics of the dierential mOD for the single quantum well m-plane InGaN sample probed for delay times as long as 950ps. 43 The exciton-exciton interactions specic to the spectral region about 390nm have been ignored [CKM + 89]. E e ' 2jE L j 2 (5.4) Where E e , , E L , and are the dynamic Stark shift in the exciton resonance, the dipole matrix element, the electric eld for the pump, the detuning of the pump photon energy from that of the exciton resonance. While the term 2jE L j 2 expresses the dynamic Stark shift of the s andp states that form the conduction and valence bands the second term in E e , , is the renormalization of that shift due to many body eects. In this study we extracted a value of 22.43mev for E e from the experimental data. On the other hand the dipole matrix element can be estimated using the following formula [CK99]: jj 2 = ~ 2 e 2 2m 0 E g m 0 m e 1 1 + 1 + 2 E g (5.5) Where m 0 , m e , 1 , 2 , and E g are the electron rest mass, the electron eective mass, the crystal energy splitting, the spin-orbit energy splitting, and the band gap energy. We evaluated the dipole matrix element for known values of GaN [VM03] and obtained a value ofe2:049 A. This, of course, is merely an approximation as the exact value of the dipole matrix element does depend on the In composition and connement. For a pump intensity of 1.8GW=cm 2 at 400nm, that corresponds to detuning of = 80mev, and after accounting for Fresnel re ection at the air-material interface the parameter was estimated to be 4.22. This number for GaAs/AlGaAs based material structures, being almost independent of the dimensionality of the connement, is about 3.5 and 2.285 in 44 three and two dimensions [SRCH88]. Now = jj 2 N PSF , where is the 1s exciton probability amplitude and N PSF is the Mott density. For a Mott density of 10 19 cm 3 [BDOS99] and by approximatingjj 2 asjj 2 = 1 2a 2 2D we extracted a two dimensional exciton Bohr radius, a 2D , of 0.613nm. The exciton binding energy, then, can be estimated to be 17.81mev, which compares well with numerical calculations [PKK04]. The energy gap corresponding to the exci- tonic resonance at 390nm can also be estimated. We obtained a value of 388nm, to be compared with the static photoluminescence of the target sample that exhibited an emission peak at 389nm. In conclusion we have observed a strong evidence for the dynamic Stark eect at 390nm in the transient absorption studies that were conducted on an m-plane InGaN single quantum well structure. The experimental data was then used to estimate a two dimen- sional Bohr radius of a 2D = 0:613nm, which revealed an exciton binding energy of 17.81mev and a corresponding band-edge energy of 388nm. 45 Chapter 6 Laser dynamics: Probing microscopic processes in InGaN light emitters. This chapter proposes a study that investigates the dynamics of a InGaN/GaN quantum well laser as a mean to probe the Auger scattering rate. As the carrier density clamps at the threshold, an injection optical eld that is spectrally close to the lasing frequency can modulate the carriers in the cavity. Dynamics of the lasing eld then carries infor- mation about individual scattering processes since these processes are carrier density dependent. In this section microscopic semiconductor-Bloch equations are set up to address the Auger scattering process side-by-side with the carrier-carrier and carrier-phonon scatter- ing processes. The collision relaxation rates can be better approximated by considering a total carrier density dependence. This carrier density dependence is calculated after analyzing the Coulomb correlation eects. The carriers are allowed to escape into the barriers therefore allowing the particles to be exchanged between the barriers and the quantum wells. The polarization eects and the screening of that eect can be handled through a carrier density dependent band structure that is obtained by solving the k.p- Poisson equations. The goal is to demonstrate that these equations can be used along side with a laser modulation measurement to study the eects of the Auger process on the relaxation oscillation. To achieve that, the semiconductor-Bloch equations are ana- lytically manipulated to reveal the dependence of the relaxation oscillation frequency to 46 the microscopic processes including the Auger scattering process. Under the relaxation rate approximation we start with the following equations. The denition of variables are given in table-6.1. dp i;j k dt = i ! i;j k ! p i;j k i i;j k n i e;k +n j h;k 1 p p i;j k (6.1) dn i e;k dt = X j i i;j k p i;j k +c:c: nr n i e;k cc n i e;k f i e;k cp n i e;k f i e;k (6.2) ag n i e;k f i e;k R inj +G pmp dn j h;k dt = X i i i;j k p i;j k +c:c: nr n j h;k cc n j h;k f i h;k cp n j h;k f j h;k (6.3) ag n j h;k f j h;k R inj +G pmp dn b ;k dt = b nr n b ;k b cc n b ;k f b ;k b cp n b ;k f b ;k b ag n b ;k f b ;k B k n i e;k n j h;k (6.4) de (t) dt = c e (t) + i! b V qw X i;j;k i;j k p i;j k (6.5) 47 In above equations R inj and G pmp are the carrier consumption and generation rates caused by interaction of the medium with modulating and pump elds respectively. These rates are dened as R inj = X 1 2 i;j k e inj (t) 2~ 2 L inj k i;j k is i(j) for holes(electrons) (6.6) G pmp = X 1 2 i;j k e pmp (t) 2~ 2 L pmp k i;j k is i(j) for holes(electrons) (6.7) L inj k and L inj k are the injection and pump eld lineshape functions receptively and k is the inversion given below. L pmp k = 4 pmp 2 pmp + ! i;j k ! pmp 2 L inj k = 4 inj 2 inj + ! i;j k ! inj 2 i;j k = n i e;k +n j h;k 1 i;j k and and ! i;j k are renormalized Rabi and transition frequencies and are dened as i;j k = k E (z;t) ~ + 1 ~ X k 0 6=k V jkk0j p i;j k0 (6.8) ! i;j k = !0 i;j k X k 0 6=k V jkk0j n i e;k0 +n j h;k0 (6.9) The carrier distributions of the quantum well and the bulk regions are coupled through conservation of total carrier density and the law of conservation of energy. The carriers are allowed to escape or be captured back into the quantum wells as long as they conform 48 with conservation of particle and energy laws. Because there is a particle exchange between the bulk and quantum well regions the chemical potentials are allowed to be continuous across the bulk-quantum well boundaries. The following equations can be used to extract the temperatures and chemical potentials. N = n qw A X i;k n i ;k + w b V b X k n b ;k (6.10) = n qw A X i;k f i ;k ( p ;T p ) + w b V b X k f b ;k ( p ;T p ) = n qw A X i;k n i ;k i ;k + w b V b X k n b ;k b ;k (6.11) = n qw A X i;k f i ;k ( p ;T p ) i ;k + w b V b X k f b ;k ( p ;T p ) b ;k In Eq. 6.10 and Eq. 6.11, p andT p represent the chemical potential and the temperature for carrier-carrier, carrier-phonon, and Auger scattering processes whenp = cc,p = cp, andp = ag respectively. For the carrier-phonon scattering only equation (15) is needed as one could always set the lattice temperature to be that of the environment. We will now recast the semiconductor-Bloch equations in a form that could reveal the connection between the relaxation oscillation frequency and the microscopic collision rates. For that purpose, and for the sake of simplicity, we will consider a two sub-band system and neglect the Hartree-Fock renormalization eects, but retain the collision terms under the relaxation rate approximation. We will use the equations 6.1 through 6.5 to nd rate equations for e (t) e (t) and the inversion given by i;j k =n i e;k +n j h;k 1. With the assumption that the absorption rate at the pump frequency is unaected the 49 following rate equations are derived. For a numerical calculation this assumptions can be relaxed. d (ee ) dt = 2 c (ee ) ! b V qw X k Imf e p k g (6.12) k dt = 1 ~ Imf e p k g ( k + 1) +R eq k e inj (t) 2~ 2 L inj k k (6.13) Were and R eq are given by = nr + cc + cp + ag (6.14) R eq = X p=cc;cp;ag p f e;k ( p e ;T p ) +f h;k p h ;T p (6.15) Assuming that the carrier density distributions and the eld envelope vary little in the microscopic polarization life time one can directly integrate the polarization rate equation and pull the above mentioned quantities outside of the integral (rate equation 50 approximation) further simplifying using the rotating wave approximation the slowly varying part of the polarization becomes p k = i 2~ k e k 1 i (! k !i p ) (6.16) that leads to Imf e p k g = 1 8~ j k j 2 (ee ) L p k k where L p k = 4 p 2 p + ! i;j k ! 2 (6.17) Renaming ee and e inj e inj to I and I inj , we rewrite Eq. 6.12 and Eq. 6.13 as dI dt = 2 c I ! b V qw X k n 1 8~ j k j 2 IL p k k o (6.18) d k dt = 1 8~ 2 j k j 2 IL p k k ( k + 1) +R eq k 2~ 2 I inj L inj k k (6.19) Applying the dierential operator to Eq. 6.18 and Eq. 6.19 and substituting i for d dt we have ( i + 2 c +A (!) X k p k k i + 0 k i + k ) dI = A (!) ( X k p k inj k i + k k I ) dI inj (6.20) +A (!) ( X k p k i + k I ) dR eq 51 In deriving Eq. 6.20 the following denitions have been used. Note that I have also assumed that the dipole matrix element of the injection eld is dierent from that of the laser eld. p k = 1 8~ 2 j k j 2 L p k inj k = 1 4~ 2 j inj k j 2 L inj k A (!) = ~! b V qw k = + inj k I inj + p k I 0 k = + inj k I inj The zeros of the left side of Eq. 6.20 are the poles of the modulation response of the system and describe the dynamics of the system including the relaxation oscillation frequency and the damping coecient. To better illustrate this point let us assume that there is only one dominant term in the sums and that the corresponding momentum is k 0 . Then the term containing the poles of the frequency response of the system can be written as (The dR eq term on the right side can be absorbed into the dI inj term) i k0 + 2 c A (!) p k0 k0 + 2 c k0 A (!) p k0 k0 k0 2 Let us recall that = nr + cc + cp + ag embeds the collision rates we are interested in. Now we can write the following where we have used k0 p k0 I = inj k0 I inj + ! 2 R = 2 c A (!) p k0 k0 p k0 I + inj k0 I inj + (6.21) Under relaxation rate approximation the collision rates are usually assigned with constant numbers although they in reality are carrier density and temperature dependent. Typical 52 values for the carrier-carrier and carrier-phonon collision rates are 1 to 2 10 13 s 1 and 0.2 to 1 10 12 s 1 . Ref. [CCTK10] suggests an Auger collision rate of the form C ag N 2 where N is the total carrier density. To better understand carrier density dependence of the collision it is necessary to consider the correlation eects. Let us start with the collisions term for the electrons under the carrier-carrier scattering eect, which is given by the following. @n ek @t cc = n ek out X ek fng + (1n ek ) in X ek fng (6.22) where out X ek fng = ~ X b=e;h X q6=0 X k0 2V 2 q e;b V q V jkk0+qj e;k + b;k0 e;k+q b;k0q (6.23) 1n e;k n b;k0 1n b;k0q and in X ek fng = ~ X b=e;h X q6=0 X k0 2V 2 q e;b V q V jkk0+qj e;k + b;k0 e;k+q b;k0q (6.24) n e;k+q 1n b;k0q n b;k0q Similar equations can be written for hole density distribution by interchanging e and h subscripts. To get to the relaxation rate form we write the total relaxation rate ofn ek and n h;k as a single decay rate given by cc fng = P out k fng + P in k fng. Exploiting the fact 53 that at the equilibrium the detailed balance condition prevails and that the Boltzmann scattering integral is identically zero at that condition, the following is achieved. cc ffg = ~ X q6=0 X k0 2V 2 q V q V jkk0+qj e;k + e;k0 e;k+q e;k0q (6.25) f e;k0 1f e;k+q + ~ X q6=0 X k0 2V 2 q e;k + h;k0 e;k+q h;k0q n f h;k 1f e;k+q +f h;k0q f e;k+q f h;k+q o We clearly see that under Boltzmann-Maxwell statistics that is expected at carrier den- sities around 5 10 18 cm 3 , where the eciency peaks, the carrier-carrier collision rate has a quadratic dependence on the total carrier density. This suggests that it is also possible to approximate the carrier-carrier collision rate by a term that would look like cc = C cc N 2 . However for many semiconductors the carrier-carrier scattering domi- nates the sub-picosecond regime and if we are studying a process that is much slower than the time scales associated with carrier-carrier collision rates then we can assume that an equilibrium is achieved via the carrier-carrier scattering process with its respec- tive temperature and chemical potential and eliminate that process through detailed balancing from out equations. It is now clear that, as stated before, the Auger process will have the same form of carrier density dependence. This is because the Auger process has the same interaction Hamiltonian as the carrier-carrier collisions. Note that inter- band processes have been removed from the calculation when deriving Eq. 6.23 and Eq. 6.24. Now it remains to investigate the carrier density dependence of the carrier-phonon scattering process. First we will cast the carrier-phonon collision rate in a form that will 54 resemble that of Eq. 6.22 then we will follow a similar procedure as that we used to derive Eq. 6.25. The carrier-phonon collision rate can be written as the following. @n e;k @t cp = n e;k ( 2 X q G 2 q (1n e;kq ) + k;q (n ph;+q + 1) + k;q n ph;q ! (6.26) + 2 X q G 2 q n e;kq k;q (n ph;q + 1) + k;q n ph;+q !) 2 X q G 2 q n e;kq k;q (n ph;q + 1) + k;q n ph;+q ! In Eq. 6.26 G q is the Frohlich electron-LOphonon coupling and + , + , and n ph are dened below. Also note that the rst term describes the out-scattering events while the second and third terms describe the in-scattering events. ~ + = k ( kq +~! LO ) ~ = k ( kq ~! LO ) n ph = 1 e ~! LO 1 55 While noticing that equation (31) now looks liken e;k n P out q fng+ P in q fng o + P in q fng then the term in curly brackets evaluated at the equilibrium will give the total relaxation rate due to carrier-phonon scattering as follows. cp = 2 X q G 2 q (1f e;kq ) + k;q (n ph;+q + 1) + k;q n ph;q ! (6.27) + 2 X q G 2 q f e;kq k;q (n ph;q + 1) + k;q n ph;+q ! In Eq. 6.27 tells us that the carrier-phonon collision rate can be approximated by a term that is proportional with the total carrier density. (C cp N) We employed a three dimensional FDTD code [ORI + 10] deployed on 60 dual-core pro- cessors at the HPCC of USC to design our laser cavities. Fig. 6.1 illustrates mode prole of a microdisk. Figure 6.1: Cross section views of the Ey (perpendicular to growth direction) component of the electric eld. Three modes are shown for a 10m microdisk containing 6 pairs of InGaN/GaN of quantum wells The microdisk has a thickness of 240 nm, which includes two GaN cap layers each 80 nm of thickness and a gain region that is 80nm thick embedding 6 pairs of InGaN/GaN 56 quantum wells. The microdisk sits on a 20nm SiNx layer that has been grown by PECVD on top of a Sapphire substrate that is 2m thick in this calculation. The refractive indices of the sapphire (1.77), GaN (2.5), and InGaN/GaN (2.35) quantum well layers were taken from Ref. [TASJ98], Ref. [PN02], and Ref. [LDL98] respectively. The refractive index of the SiNx (2.04) layer was calculated using the Sellmeire equation. Fig. 6.2 shows the cavity Qs for the three modes shown in the Fig.6.1. Figure 6.2: Calculated cavity Q for the three modes of the microdisk shown in Fig.6.1 Microdisk lasers with a similar composition for the gain region with Q-s of about 5000 were demonstrated [HSM + 04] under pulsed condition. CW microdisk lasers (Q of 3500) with InGaN/GaN quantum wells have also been demonstrated [THS + 06]. We think that the cavity Q for our design is high enough to achieve lasing. We have also investigated photonic crystal cavities. The extent of freedom in design of the mode structure and dispersion of these cavities make it possible for the optical mode to be coupled to a photonic crystal waveguide and in-principle, almost entirely, collected via a tapered optical ber. Fig.6.3 depicts the optical mode of a GaN membrane photonic crystal cavity (L7) brought to the proximity of a GaN substrate. This mode has a center wavelength of 475nm and the optical spectrum of the cavity can be seen in Fig.6.4. As the distance between the cavity and the substrate becomes smaller and smaller the mode leaks into the substrate and the Q of the cavity drops dramatically as illustrated in Fig.6.5. 57 Figure 6.3: Cross section views of the Ey (perpendicular to growth direction) component of the electric eld. The membrane is 240nm thick and the gap between the membrane and the substrate is 320nm Figure 6.4: Calculated optical spectrum of a GaN membrane L7 cavity in the proximity of a substrate with a gap of 160nm Although there is a strong evidence for the two strongly debated causes of the e- ciency droop, namely the Auger recombination and the carrier escape, a need for an experiment that relies on a microscopic theory is eminent. We mentioned earlier that as the carrier density clamps at the threshold, an injection optical eld that is spectrally close to the lasing frequency can modulate the carriers in the cavity. Dynamics of the lasing eld then carries information about individual scattering processes since these processes are carrier density dependent To further clarify this issue let us start with the 58 Figure 6.5: Calculated cavity Q for the mode of the L7 cavity at the center wavelength of 475nm versus the distance of the membrane and the substrate. logarithmic gain model and estimate the ratio of total carrier densities for two cavities identied with quality factors of Q1 and Q2. It is possible to show the following. N th 1 N th 2 = exp 2n g g 0 Q 2 Q 1 Q 1 Q 2 ! (6.28) In the Eq. 6.28 N th , n g , , , g 0 , and Q are the threshold carrier density, the group index, the wavelength, the connement factor, the empirical gain coecient and the cavity Q respectively. For Q 1 = 4000, Q 2 = 6000, = 405 [nm], n g = 3:8 [SSS + 11], = 0:02, andg 0 = 5000[cm 1 ] the fraction of the carrier densities at threshold becomes N th 1 N th 2 = 1:6. This means that there will be approximately a 2.57 fold dierence between the corresponding ag -s, since ag = C ag N 2 th . We also assume that the C ag does not change from one cavity to the other, which is a reasonable assumption if the cavities are fabricated under the same lithography window. It has been shown, by numerical calculation [CSK + 02], that a signicant change in the relaxation oscillation frequency (20 GHz) is expected as the the carrier-phonon collision relaxation rate is changed ve folds (from 10 12 s 1 to 2 10 11 s 1 ) as can be seen in Fig.6.6 59 Figure 6.6: Calculated modulation response of a VCSEL [CSK + 02]. Although a very high ! R has been predicted [CSK + 02] a relatively low relaxation oscillation frequency of 4.7 to 7.3 GHz has been experimentally demonstrated in Nitride- based blue lasers [MSS + 10], as illustrated in Fig.6.7. The smaller relaxation oscillation frequencies have been attributed to higher threshold carrier densities compared to GaAs and InP based devices. Figure 6.7: Modulation response of a Nitride-based laser emitting at 450nm [MSS + 10]. In an electrically addressed Nitride-based laser most of the heat comes from joule heating due to high resistivity of p-doped region. The much smaller contribution is from non-radiative defect recombination, which is partly compensated by the phonon absorption through thermionic escape of the carriers from the quantum wells [PN02]. Although if we optically excite the carriers within the quantum wells the joule heating 60 and carrier yover do not apply. As stated before the carrier escape is taken into account since Eq. 6.10 and Eq. 6.11 couple the three dimensional carrier densities to the two dimensional carrier densities of the quantum wells. These equations also are used to calculated the respective temperatures of the dierent microscopic processes. In conclusion, we have investigated the lasing dynamics for lasers that employ InGaN- based quantum wells as their gain medium and proposed an experiment that would allow extraction of microscopic parameters. We also deployed FDTD calculation schemes to design the laser cavities for this purpose. 61 Table 6.1: Denition of the variables in the semiconductor-Bloch equations Variable Denition k 2D quantum well carrier momentum and the spin of the carriers i,j electron or hole sub-band numbers c cavity line-width inj injection eld line-width pmp pump eld line-width nr SRH non-radiative recombination rate coecient cc carrier-carrier collision rate cp carrier-photon collision rate p polarization dephasing rate ag Auger collision rate ! laser eld frequency e (t) slowly varying complex electric eld b background permittivity optical mode connement factor i;j k dipole matrix element p i;j k microscopic polarization n i e;k electron distribution f i e;k electron Fermi-Dirac distribution n j h;k hole distribution f j h;k hole Fermi-Dirac distribution n b ;k bulk electron or hole distribution f b ;k bulk Fermi-Dirac electron electron or hole distribution B k spontaneous emission rate coecient V jkk0j Fourier transform of Coulomb interaction potential V qw volume of the gain region n qw number of quantum wells A area of the quantum wells w b thickness of the barrier region V b volume of the barrier region T cc temperature of the carriers T cp lattice temperature T ag temperature of the Auger process cc chemical potential associated with the carrier-carrier scattering cp chemical potential associated with the carrier-phonon scattering ag chemical potential associated with the Auger process i ;k quantum well electron or hole energy b ;k bulk electron or hole energy total energy density N total carrier density of electrons or holes 62 Chapter 7 Photonic crystals: A short introduction Photonic crystals are an optical analogue of electronic crystals where periodic arrange- ment of atoms alter the electronic properties in a very intriguing manner. Just as elec- tronic crystals provide a pathway for electronic propagation without scattering, the opti- cal crystals do the same for photons. On the other hand this periodicity of electronic or optical media can also prohibit the propagation of light or electrons for certain ener- gies. This is possible when the photonic crystal possesses an energy band gap just like their electronic counterparts. Photonic crystals are made by macroscopically altering the dielectric constant of the medium where light propagates. This may be done in one, two, or three dimensions. For example a periodic stack of two dierent materials with alter- nating refractive indices would constitute a one dimensional photonic crystal. As one may expect one dimensional photonic crystals could inhibit the light propagation in one dimension while two and three dimensional photonic crystals can inhibit light propaga- tion in two and three dimensions. The mechanism behind this inhibition of propagation lies in the fact that the forward and backward propagating waves that would naturally be independently propagating inside the unaltered medium couple their energy to each other when the periodicity is introduced. Let us now imagine a light emitter, like an excited atom, is placed in a photonic crystal. The radiating atom could couple its light into an optical mode propagating towards a certain direction where the photonic crys- tal has been engineered to possess an energy gap. That optical mode after a certain propagation length will couple its energy into the backward propagating mode relaying 63 the energy back into the photonic crystal. This means that photonic crystals can be engineered for an optical resonator. Very high optical quality factors have already been demonstrated for photonic crystal cavities [AASN03]. A photonic crystal can be altered to guide the light towards a certain optical crystallographic direction. Such devices are known as photonic crystal waveguides [JVF97a]. Fig. 7.1 embeds a movie illustrating an optical mode in a photonic crystal waveguide resonator. The optical cavity is formed by prohibiting the light propagation in the planar direction by introducing periodicity in a slab of high index material suspended in air. The periodicity is introduced in the form of air cylinders drilled into the material (Si in this example). Additionally, the contrast of refractive indices in the vertical direction provides internal re ection. The light is allowed to propagate in the waveguide direction by removing a row of holes. It is interesting to note that for this optical mode the light does not need to reach the facets before being redirected into the cavity in the direction of propagation. Implementation of a photonic crystal technology platform for on-chip (Loading playme.mpeg) Figure 7.1: Please double click on the image to play. This movie was produced by performing an FDTD calculation on USC's High-Performance Computing and Commu- nications (HPCC) super computer. photonic communication has been suggested and there is a signicant body of work in 64 literature on this topic [Kra01]. However, in this introduction I would like to stress the fact that the photonic crystal platform provides an opportunity to control the ow of light by altering the periodicity. The pitch of this periodicity is determined by the energy or frequency of the propagating photons. The frequency of the propagating light is set by the material embodying the photonic crystal and is naturally chosen such that there are no material absorption, except of course for regions where the light itself is generated. Depending on the application the light may be coupled to the photonic crystal device from outside. If the frequency of propagating light is relatively high the feature size of the photonic crystal device becomes inevitably small. This poses a problem for coupling the light inside the device or devices. The output light of the optical bers diverges very quickly and the dimension of it for on-chip communication applications is about 2m (with a focusing lens in place). The thickness of the guiding layer for a waveguide designed for that application is about 240nm. This means that most of the input light will go waste by freely ying around the waveguide leaving only a tiny portion to couple into the waveguide. A solution to this problem is evanescent coupling. Chapter 8 of this manuscript discusses a technology that enables the researcher to build a light coupler that is laboratory robust and possibly implementable in commercial applications. This device takes advantage of evanescent coupling between a tapered ber and a photonic crystal waveguide resonator. The advance of this work is the fact that the ber is directly positioned on top of the waveguide region and is held down by microscopic forces such as hydrophilic, electrostatic, and Van Der Waals forces. The technology used to fabricate this device enables us to build the device without a need for a controlled environment. Typical laboratory settings are such that the ber is brought to the proximity of the waveguide for purposes of light coupling, which means that even very small movements of the ber taper (less than1 m in diameter) will aect the coupling. We mentioned earlier that the periodicity of refractive index in photonic crystal devices can be used to redirect all light towards a certain conned region to form an optical cav- ity or it can be manipulated to channel the light towards a desired direction to form a 65 waveguide. This redirection of light using periodic structures not only aects the ampli- tude of redirected light but is also alters the phase. This is because the light needs a little time to traverse inside the periodic medium before it can couple its energy to the redirected light. This time delay contributed by photonic crystal devices is the subject of the investigation in chapter 9. In that chapter a theory of photonic crystal waveguide resonators is proposed that enables extraction of phase contributions of a photonic crys- tal waveguide resonator. The light coupler discussed in chapter 8 is fabricated in such a way that all four ports of it are accessible for light injection. The two facets of the photonic crystal waveguide resonator can be accessed by launching a laser light in free space that is directed and eventually focused on the input facet of the resonator while the other facet can be used for out-coupling and detection. The tapered ber can be used to inject another laser into the device. This provides an opportunity for investigation of simultaneous interaction of dierent light sources inside the photonic crystal cavity. Because the light can be coupled very eciently via the ber taper it is possible to study the eects of a very intense light on the transmission of weak laser light that is coupled and detected from the facets of the photonic crystal waveguide. As discussed in chapter 10 this device has been used to demonstrate all-optical nonlinear shift and bleaching of cavity resonances via non-degenerate two photon absorption, non-degenerate Kerr mechanism, free carrier absorption, and free carrier plasma eects. As the response time of two photon pro- cesses are very fast, about 10 fs, this device can be used in ultrafast low energy all optical switching applications. Furthermore, it is possible to extract such interesting physical parameters as non-degenerate two photon absorption and non-degenerate Kerr coecients. 66 Chapter 8 An ecient four-port photonic crystal ber taper coupler Planar photonic crystal technology has undergone extensive research as a technology platform for integration of optical elements [JVF97b, Kra03]. One aspect of this tech- nology is to successfully interface an integrated optical circuit with the outside world. To address this issue, coupling between a tapered ber and photonic crystal waveg- uides have been demonstrated. Both theoretical [KKSO02, BSP03] and experimental [BSBP04] studies have been carried out that conrm an ecient interfacing between an integrated circuit and outside environment is possible. Optimization of the coupling has been investigated [HKL06] and an enhanced probing of the resonances of a photonic crys- tal Fabry-Perot waveguide cavity for the purpose of characterizing the photonic crystal waveguides via a curved tapered ber has been reported [LGP + 08]. In this work we also probe the resonances of a photonic crystal Fabry-Perot resonator and it is done based on a full four-port modeling of the coupler where the coupling junction is considered to change both the amplitude and phase of the elds. The spectroscopic technique devel- oped throughout this chapter enables us to characterize the line shape of the resonances beyond that of a simple Fabry-Perot cavity. The model can be benecial when four-port measurements are to be conducted or direct extraction of the cavity Q is not readily possible (due to broad resonances for instance). The coupler is fabricated by directly sit- uating the tapered ber on a silicon membrane photonic crystal waveguide. The tapered ber partially overlaps with the defect region of the waveguide and the waveguide is lithographically terminated resulting in a Fabry-Perot cavity. 67 Fig. 8.1 illustrates the technique we used in the fabrication of the coupler. As can be seen from this gure the rst stage is the process of tapering which is done by heating and simultaneously pulling the ber until a diameters of about 1m was reached. The heat zone is created by a spatially oscillating gas burner torch that utilizes a mixture of Hydrogen and Oxygen gases. The torch and the ber pulling stages are all motorized and must be carefully controlled to achieve adiabatic tapering. Theoretical and experimental investigation of adiabatic tapering of optical bers have been thoroughly studied before [LHS + 91, BLGL91]. The second step is to drive back the motorized stages along the z axis while they are also moved away from each other along the y axis. This allows for the tapered region to be positioned along the defect region of the waveguide. The third process is to raise the motorized stage carrying the photonic crystal waveguide to the proximity of the ber where the ber is then accurately situated on the photonic crystal waveguide defect. We contrast this process with previously reported techniques where x z y Tapering Reshaping Motorized Stage Tapered Fiber Figure 8.1: An illustration of the technique used in fabrication of the photonic coupler under study the ber is suspended in the air and a post processing is necessary to mount the ber on a xture. Note that the reshaping of the ber is done in room temperature and can be very quick utilizing the motorized stages. After positioning, the ber stays in place for days and no glue or such substances were used. The ber may be recongured to 68 meet a particular purpose. One example of this reconguration is achieved by moving the stages in such manners that curve the ber away from the defect region near the facets (see Fig. 8.2) enabling the eective length of the coupler to be adjusted. This also helps in conducting free space measurements as the far eld distribution of the light emanating from the facets is not disturbed. Placing the ber on the device can cause the light to couple to the top silicon slab and scatter at the point of entry which will reduce the amount of light available for input-output coupling. This problem can be partially addressed by increasing the number of photonic crystal periods at the two sides of the defect but the scattering of the light will persist. The couplers made by directly situating the ber on the waveguide may have limited coupling band width compared to those utilizing an extended k-space coupling scheme [LGP + 08]. One major challenge in achieving a controlled positioning of the ber on the defect region of the waveguide is the air ow and vibration of the ber as the stages move. We think that a controlled environment can alleviate this problem. Fig. 8.2 displays the actual fabricated device Figure 8.2: Top microscope view of tapered ber partially overlapping the waveguide defect region. utilizing the technology described here. As can be seen in this gure and more clearly in Fig. 8.3, the waveguide is lithographically terminated at both sides to allow free space probing of the light. The waveguide is 150m in length and the lattice constant, hole radius, membrane thickness and the separation between the membrane and the substrate are 420nm, 120nm, 250nm, and 2m respectively. This waveguide has a transmission window between 1540nm and 1620nm. Note that this technique allows for integration of the entire setup, including the free space optics, and that no transportation of the tapered ber is necessary. Therefore the tapered ber fabrication, interfacing with a 69 photonic crystal circuit, and full four-port measurements all are possible in one place. We will now derive the transfer functions for all four ports of the coupler using a matrix Figure 8.3: SEM image of a fabricated photonic crystal waveguide with lithographically dened facets. method [San91]. Fig. 8.4 illustrates our four-port device. We assume that the coupler is lossless and instead take all the losses to arise from the coupler's arms. We also assume identical facet re ectivities considering the fact that the facets have identical termina- tion of photonic crystal lattice. Assuming a contradirectional coupling scheme we start setting up our equations for the case where the input is at the left facet of the waveguide. (See sets of Eq. (8.1)) [κ] Ea1 P1 P2 P3 P4 Ef1 Ea1 Ef1 Ef3 Ea3 Ef4 Ea4 Ein r2 r1 Eout1 Eout2 Eout3 Eout4 Figure 8.4: An illustration of the coupler. The black lines represent the ber and the red lines represent the waveguide 70 E a = []E f ; E f = [R]E a + [t in ]E in ; E out = [T ] (I [] [R]) 1 [] [t in ] E in [] = 0 B B B B B B B @ 0 p 12 0 p 14 p 21 0 p 23 0 0 p 23 0 p 21 p 14 0 p 12 0 1 C C C C C C C A ; [R] = 0 B B B B B B B @ re 2(+ig )lg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 re 2(+ig )lg 1 C C C C C C C A [t in ] = 0 B B B B B B B @ p in e 2(+ig )lg 0 0 0 1 C C C C C C C A ; [T ] = 0 B B B B B B B @ te (+ig )lg 0 0 0 0 e i f l f 0 0 0 0 e i f l f 0 0 0 0 te (+ig )lg 1 C C C C C C C A (8.1) Here all the parameters in [] are complex. is the loss associated with the waveguide, and f and g are the ber and waveguide propagation constants. l g and l f are the lengths of the coupler's arms for waveguide and ber respectively. r, t, and p in are re ectivity, transmissivity, and coupling coecient of the waveguide facets. Solving for the transfer function, [T ] (I [] [R]) 1 [] [t in ] yields: 0 B B B B B B B B @ e 3lg (+ig) rt p in 14 1e 4lg (+ig) r 2 14 e 4lg (+ig ) p 21 e i f l f + e i f l f 4lg (+ig) r 2 p 21 14 1e 4lg (+ig) r 2 14 p in e i f l f 3lg (+ig) r p 21 p 14 p in 1e 4lg (+ig) r 2 14 e 2lg (+ig) t p 14 p in 1e 4lg (+ig) r 2 14 1 C C C C C C C C A (8.2) 71 Had we assigned the input to the port 3 in Fig. 8.4 we would have obtained: 0 B B B B B B B B B @ e i f l f 3lg (+ig) rt p 12 p 14 1e 4lg (+ig) r 2 14 e i f l f p 23 e i f l f + e i f l f 4lg (+ig) r 2 p 21 p 12 p 14 1e 4lg (+ig) r 2 14 e 2i f l f 2lg (+ig) r p 21 p 12 1e 4lg (+ig) r 2 14 e i f l f lg (+ig) t p 12 1e 4lg (+ig) r 2 p 14 1 C C C C C C C C C A (8.3) Each term in these matrices can be interpreted. For example the second element in matrix (8.2) shows that the light emanating from the ber at port 2 is a combination of the light that is contradirectionally coupled from port 1 and the light that is con- tradirectionally and repeatedly coupled from the port 1 but after one complete round trip along the waveguide and through the coupler. Now let's take the fourth element of matrix (8.2), by setting p 14 = 1 andL g = 2l g one recovers the transfer function of the unloaded waveguide resonator where L g is the length of the waveguide resonator. To develop the spectroscopy tools for analyzing our experimental data we will rst start with our unloaded (before the ber is placed on defect region of the waveguide) Fabry- Perot cavity. The output power detected o of the right facet can be written as: I out = t 2 e 2Lg in 1 +R 2 2R cos (x) I in (8.4) Where R =r 2 e 2Lg and x = 2 g L g . Following Ref. [Coo71], the transfer function in Eq. (9.1) can be recast in the following form (see Eq. [8-10]): 2 p 2t 2 e 2Lg in 1R 2 lnR x 2 + (lnR) 2 +1 X q =1 (2qx) (8.5) Fig. 8.5 reveals that the spectrum of a Fabry-Perot cavity can be described by con- volution of Dirac-Comb function and a broadening function that has a Lorentzian line shape. After performing the convolution it can be shown that the transmission spectrum of the waveguide can be t by Lorentzians where the width of the Lorentzian have the 72 following form HWHM = 1 ng ln 1 R where n g is the group index. The group index 1180 1190 1200 1210 1220 4 Π €€€€€€€€ €€ € Λ L 2 4 6 8 Output Power @ΜWD The Fit Detected power Figure 8.5: Transmission spectrum of the unloaded waveguide and the t obtained via expansion by Lorentzians can be extracted from the relative positions of the Lorentzians therefore the loss can be extracted from the width of the Lorentzians. Fig. 8.5 depicts the detected power from the right facet of the unloaded waveguide, when the light was launched through its left facet, and the t to the data using the above technique. Fig. 8.6 shows the extracted FWHM and the group index. The minimum waveguide propagation loss we have measured for this waveguide, using this technique, is 4:7db=mm ( A mirror re ectivity of 0.4 was assumed). The choice of 4L for the horizantal axes becomes apparent after performing the convolution in Eq. (8.5) and regrouping the terms. With this choice of units the spacing between the Lorentzians is inversely proportional to the group index with a factor of 2. We have compared this technique with the more generally used Hakki-Paoli method [HP75] for both experimen- tal data and that of an articially generated data, and the two methods are in very good agreement. Characterizing photonic crystal waveguides using Fabry-Perot resonators has been done before [NYS + 01]. Direct measurement of the group index of photonic crystal waveguides has also been reported [GIOO + 07]. Our method can be benecial for the instances where there is not enough spectral resolution to condently determine the values and spectral positions of the peaks and valleys. Resolving all the resonances in 73 1180 1190 1200 1210 1220 4 Π €€€€€€€€ €€ Λ L 5 10 15 20 25 Group Index 1180 1190 1200 1210 1220 4 Π €€€€€€€€ €€ Λ L 1 2 3 4 5 FWHM Figure 8.6: FWHM and group index extracted by tting the spectrum of the unloaded waveguide resonator the lab can be a very lengthy process and it may take hours to scan the entire spectrum where the optics can signicantly shift during the measurements. Going back to the fourth element of the matrix 8.2 the output power emitting from the right facet can be written as: I out = t 2 e 2Lg j 14 j in 1 +R 2 2R cos (x + 14 ) I in (8.6) Where R =r 2 e 2Lg j 14 j, x = 2 g L g , and 14 =j 14 je i 14 . Let us dene H (x) = 1 1+R 2 2R cos(x+ 14 ) and expand H (x) in powers of 14 so that 74 H (x) = A 0 +A 1 14 +A 2 2 14 + . Comparing the expansion coecients reveals that they have the following regenerative form: A 0 = 1 1 +R 2 2R cos (x) A 1 = 2R cos x + 2 1! (1 +R 2 2R cos (x)) A 0 A 2 = 2R cos x + 2 2 2! (1 +R 2 2R cos (x)) A 0 + 2R cos x + 2 1! (1 +R 2 2R cos (x)) A 1 A 3 = 2R cos x + 3 2 3! (1 +R 2 2R cos (x)) A 0 + 2R cos x + 2 2 2! (1 +R 2 2R cos (x)) A 1 + 2R cos x + 2 1! (1 +R 2 2R cos (x)) A 2 : : : (8.7) Ignoring all the terms with higher powers than one of quantity R in the numerator and by substituting the coecients back in the expanded version of H (x) one realizes that H (x) can be rewritten as: H (x) = 1 1 +R 2 2R cos (x) 2R sin (x) (1 +R 2 2R cos (x)) 2 14 1! 3 14 3! + 5 14 5! +2R cos (x) (1 +R 2 2R cos (x)) 2 2 14 2! + 4 14 4! 6 14 6! (8.8) 75 The terms involving 14 can be summed up to produce, sin ( 14 ), and (cos ( 14 ) 1) respectively. Now let's turn our attention to the terms multiplying sin ( 14 ), (cos ( 14 ) 1), and A 0 . We can write by expanding to geometric Fourier series: A 0 = 1 1R 2 1 + 2R cos (x) + 2R 2 cos (2x) + 2R 3 cos (3x) + sin (x) (1 +R 2 2R cos (x)) 2 = 0 + 1 1R 2 sin (x) + 2R 1R 2 sin (2x) + 3R 2 1R 2 sin (3x) cos (x) (1 +R 2 2R cos (x)) 2 = 2R (1R 2 ) 3 + 1 1R 4 + 4R 2 (1R 2 ) 3 cos (x) + 2 1R 4 + 4R 2 (1R 2 ) 3 R cos (2x) + (8.9) Substituting Eq. (8.9) back into H (x) and taking an inverse Fourier transform to the conjugate coordinate space of x (~ x) yields (after regrouping): h (~ x) = p 2 1 1R 2 4R 2 (1 cos( 14 )) (1R 2 ) 3 R j~ xj +1 X n=1 (~ xn) p 2 1R 4 (1 cos( 14 )) (1R 2 ) 3 j~ xjR j~ xj +1 X n=1 (~ xn) p 2 i sin(~ x) 1R 2 ~ xR j~ xj +1 X n=1 (~ xn) (8.10) Now we can map h (~ x) back to H (x) and obtain: 0 @ 1 [R; 14 ] 2 ln 1 R x 2 + ln 1 R 2 ! + 2 [R; 14 ] 2 ln 1 R x 2 + ln 1 R 2 ! 2 + 3 [R; 14 ] @ @x 2 ln 1 R x 2 + ln 1 R ! 1 A p 2 +1 X q=1 (2qx) ! (8.11) 76 where, 1 [R; 14 ] = 1 1R 2 1 cos( 14 ) (1R 2 ) 3 4R 2 + 1R 4 ln 1 R ! ; 2 [R; 14 ] = 1 cos( 14 ) (1R 2 ) 3 1R 4 3 [R; 14 ] = sin( 14 ) 1R 2 Fig. 8.11 contains all the spectroscopic information we need. It tells us how the resonances of the Fabry-Perot cavity change when the tapered ber is present. As can be seen the Dirac-Comb function no longer can be assumed to be broadened by a simple Lorentzian. Instead the square of a Lorentzian and its derivative also take part. HWHM of the Lorentzians in the new line shape function are wider by 1 ng ln 1 j 14 j when compared to that of unloaded waveguide, since ln 1 r 2 e 2Lg j 14 j = ln 1 r 2 e 2Lg + ln 1 j 14 j . Other tting parameters like the area under Lorentzians and their spectral position can be taken from the t that is previously obtained from the spectrum of the unloaded waveguide. The quantity R can be recast in terms of the width of the Lorentzians. This leavesj 14 j, and 14 the only tting parameters to be obtained from the measurements performed on the coupler. Note that no assumptions are made in regard to re ectivity or losses of the facets and of the waveguide (except for that we assumed identical re ectivities for the waveguide facets), and that all the physics of the coupling mechanism is embodied in the matrix []. To further relate the elements of that matrix to such parameters as the overlap integrals and phase matching of the elds coupled mode equations have to bo solved with proper boundary conditions dictated by the coupling scheme [Oka10, BSP03]. Measurements were made by launching a free space laser beam with wavelengths within 1540nm 1620nm through the left facet of the waveguide and the output power was detected for all other ports.Knowing how the resonances broaden enables us to properly t the output power measured from all ports of the coupler. The results can be seen in Fig. 8.7 and Fig. 8.8. 77 The sources of uncertainty in obtaining the t lie in the following facts. Firstly, in deriv- ing the nal results we have made an approximation by neglecting some of the terms shown in Eq. (8.7). As the magnitude ofR becomes comparable to unity other terms in Eq. (8.7) must be included and can not be neglected. As the peak to valley ratio in the transmission spectrum becomes larger and larger the approximation becomes worse and will be more sensitive to the spectral position. In that case other terms can be included to obtain a better model. One expects that the current model works best for the spec- tral regions where the waveguide resonator is loaded the most or coupling eciency is the highest. The highest value of R reaches to 0:0175 at 1126 on the horizontal axis. Secondly, as the tting line-shapes broaden the quality of the t becomes very sensitive to the number of tting entities that are present at the periphery of any point in the spectrum. This eect should be more visible at the two ends of the spectrum. Thirdly, the model does not include the modal re ections that may exist due to non-adiabaticity of the taper. These re ections may appear as resonances in the spectrum specially when port 2 and port 3 are measured. Port 1 and port 4 are to some extent immune to this problem since the re ected power in the tapered region must couple back to the waveg- uide before it can nd its way to the facet. The coupling eciency was extracted from 1180 1190 1200 1210 1220 4 Π €€€€€€€€ €€ € Λ L 0.5 1 1.5 2 2.5 3 3.5 4 Power @ΜWD The Fit Detected power Figure 8.7: Measured output power from right waveguide facet when the input light is launched through the left facet. the Fig. 8.7 and can be seen in Fig. 8.9. It is apparent that the coupling eciency 78 1180 1190 1200 1210 1220 4 Π €€€€€€€€ €€ € Λ L 0.002 0.004 0.006 0.008 0.01 0.012 Power @ΜWD The Fit Detected power Figure 8.8: Measured output power from the right waveguide facet when the input light is launched via port 3 spectrally matches with the measured optical power from port 2. An examination of the extracted coupling eciency and the t reveals that the coupling eciency reaches to 97:3% at its highest point and the coupling FWHM is about 16:9nm. In summary we 1180 1190 1200 1210 0.000 0.005 0.010 0.015 0.020 0.025 1180 1190 1200 1210 0 20 40 60 80 100 Detected power from port 2 Coupling Efficiency[%] Output Power [ W] (4 / ) L Figure 8.9: Coupling eciency and the measured output power from port 2 when the input light is launched through the left facet discussed the fabrication technique used in making of the photonic coupler under study and presented the possible benets and drawbacks. We also demonstrated a spectro- scopic technique based on full four-port modeling and measurement and showed that by 79 knowing how the resonances of the waveguide resonator, embedded in our photonic cou- pler, are changed it is possible to extract the coupling eciency for the entire spectrum and that no assumptions are necessary for the values of the losses and facet re ectivities of the waveguide resonator. 80 Chapter 9 Group index oscillations in photonic crystal waveguide resonators Interfacing the photonic crystal devices requires a good knowledge of not only how the intensity of light changes as it couples in and out of the photonic crystal device but also knowledge of the change in phase of the electrical eld is of great importance. In this chapter a mathematical model is develop to address this issue by examining the resonances of a photonic crystal waveguide cavity. The waveguide structure has been discussed in detail in chapter 7 and will not be repeated here. The photonic crystal waveguide resonator can be considered as a Fabry-Perot cavity with mirrors that have complex re ectivities. Starting with setting up the transmission intensity of a Fabry-Perot cavity we write the following: I out = t 2 e 2Lg in 1 +R 2 2R cos (x +) I in (9.1) WhereR =r 2 e 2Lg andx = 2 g L g . is the loss associated with the waveguide, and f and g are the ber and waveguide propagation constants. L g is the waveguide length. r and t are re ectivity and transmissivity of the waveguide facets. We can expand the transmission function of the waveguide resonator using the following prescription. F (x) = a 0 + X a n cos [n (x +)] + X b n sin [n (x +)] (9.2) 81 After setting H (x) = 1 1+R 2 2R cos(x+) and using the following: a n = 1 Z + + H (x) cos [n (x +)] (9.3) b n = 1 Z + + H (x) sin [n (x +)] (9.4) a 0 = 1 Z + + H (x) (9.5) H (x) becomes: H (x) = 1 1R 2 + R 1R 2 e i(x+) +e i(x+) + R 2 1R 2 e i2(x+) +e i2(x+) +::: (9.6) Taking the Fourier transform ofH (x) into the conjugate coordinate space ofx, ~ x, results in: h (~ x) = p 2 1R 2 h (~ x) +R (~ x + 1)e i + (~ x 1)e i i + (9.7) p 2 1R 2 h R 2 (~ x + 2)e i2 + (~ x 2)e i2 i +::: We can now right a Fourier series for h (~ x) as follows: h (~ x) = p 2 1R 2 +1 X n=1 R jnj (~ xn)e in (9.8) p 2 1R 2 R j~ xj e i ~ x +1 X n=1 (~ xn) p 2 1R 2 R j~ xj e i ~ x +1 X q=1 e i2q~ x Transforming h (~ x) back to x space we obtain: H (x) = 2 p 2 1R 2 ln (R) (x +) 2 + ln (R) 2 +1 X q=1 (2qx) (9.9) 82 This means that the transmission spectrum of a photonic crystal waveguide resonator can be described by the convolution of the Dirac-comb function and a Lorentzian line- shape function that is displaced due to phase contributions of the periodicity in the architecture of the optical cavity. After performing the convolution it can be shown that the transmission spectrum of the waveguide can be t by Lorentzians where the width of the Lorentzian have the following form HWHM = 1 ng ln 1 R where n g is the group index. The group index can be extracted from the relative positions of the Lorentzians. Therefore the loss can be extracted from the width of the Lorentzians. The choice of 4L for the horizantal axes becomes apparent after performing the convolution in Eq. (9.9) and regrouping the terms. With this choice of units the spacing between the Lorentzians is inversely proportional to the group index with a factor of 2. Photonic crystal waveguide dispersion calculations predict group indices that smoothly Figure 9.1: The experimentally extracted and average group indices rises as the frequency of the light inside the waveguide approaches the photonic crystal dispersion band edge. The experimentally extracted group indices, that rely on transmis- sion measurements using a Fabry-Perot photonic crystal waveguide resonator, however exhibit oscillations. The theory presented above explains these oscillations and provides means to extract phase contribution of the optical feedback of the cavity. That is pos- sible by associating the dierence between the extracted and average group indices to 83 phase contributions. That is = (n gex n gav ) x. Where n gex and n gav are the extracted group index and the average group index. Also x is dened by x = 4 L g . Fig 9.1 depicts the group index oscillations for a waveguide resonator that is 150m long. The lattice constant, hole radius, membrane thickness and the separation between the membrane and the substrate are 420nm, 120nm, 250nm, and 2m respectively. This photonic crystal waveguide resonator has a transmission window between 1540nm and 1620nm. Fig, 9.2 illustrates the extracted phase using above theory. Figure 9.2: Extracted phase contributions of a photonic crystal waveguide resonator 84 Chapter 10 Nonliear switching of optical resonances This chapter presents an experimental study investigating an all optical tuning and bleaching of resonances for a photonic crystal waveguide resonator via nonlinear inter- action. The architecture of the photonic crystal waveguide ber taper coupler described in chapter 7 provides an opportunity for coupling laser light both through the facets (probe) of waveguide via free space and the ber taper (pump). The pump light is tuned at 1600nm and is amplied using an Erbium doped ber amplier (EDFA). The pump power is controlled either by electronically addressing the EDFA or using a variable attenuator. The probe is scanned between 1570nm and 1580nm. The resonance of the optical cavity in the heart of this device is investigated by detecting the transmission of the waveguide after interaction with an already present large reservoir of pump photons deposited via the tapered ber coupler. The pump and probe lights are then spatially ltered. This step is necessary as only a very small fraction of the probe light can be coupled into the waveguide. This is because the dimensions of the probe light (2 m in diameter) is much larger than the input facet of the photonic crystal waveguide. The light at the output port of the waveguide is imaged using an IR camera, which helps in identifying the probe light from the ambient uncoupled probe light that is also present at the camera. Having access to coupling the light using the tapered ber helps with this process. After proper ltering of the probe light it is then collected using a free-space- to-ber coupler with an eciency of about 90 %. The probe light is then directed to an optical spectrum analyzer (OSA), which is used in monochromator mode to lter the 85 probe. The probe is then detected at the detector. To make sure that power levels do not change during the course of a measurements they are electronically monitored. A data point is taken only if both the probe and pump power levels are within the tolerance set by the user. The computer program controlling the setup is self-written and is capable of controlling the measurement entirely from start to nish. The program also allows for polarization controlled measurements for all 4 directions. However, in this study we have set the polarizations of the pump and the probe for maximum transmission. OSA is also electronically addressed and the center frequency of the lter can be automatically detected. The width of the lter is set by minimum resolution of OSA and is 10nm. The Figure 10.1: Probe transmission for dierent estimated coupled pump energy at 1600 nm. setup is capable of conducting multiple measurements in which the frequency scan range and the center frequency of the lter may vary to capture the whole optical spectrum of the device. In that case the edge wavelength of the lter would need to be set such that it blocks the pump wavelength. The transmission pump power through the ber is also recorded simultaneously to monitor the possible changes in the coupling eciency of the device due to changes in the refractive index of the waveguide's body, which is made of Si. The recorded power at the output port of the ber showed a linear dependence on the input pump power conrming that there are no signicant changes in the coupling eciency. 86 Fig. 10.1 depicts the optical resonances of the photonic crystal waveguide cavity probed between 1572 nm and 1579 nm for dierent coupled pump energies indicated in the legend. As can be seen, the resonances redshift and decrease in amplitude for increased coupled optical energies for the pump light. In order to better understand the results we develop a similar theory to that presented in [USAN06]. Third order nonlinear eects in Si are described by 3 ijkl , which is the third order susceptibility tensor. 3 ijkl is the small- est non-zero nonlienar susceptibility tensor as Si is centro-symmetric. In this study we ignore the Raman contribution to 3 ijkl and only consider electronic eects as the probe photon energies are below the pump photon energy. Si possesses an m3m symmetry, which make 3 ijkl to only have four dominant components as seen in Eq. 10.1 [LPA07]. 3 ijkl = 1122 ij kl + 1212 ik jl + 1221 il jk + d ijkl (10.1) Here d = 1111 + 1122 + 1212 + 1221 and represents the nonlinear anisotropy. For degenerate two photon absorption 1122 = 1221 and for photon energies much smaller than that of the indirect band gap one can write 1212 1122 . As a result Eq. 10.1 can be written as follows: 3 ijkl = 1111 3 ( ij kl + ik jl + il jk) + (1) ijkl (10.2) The real and imaginary parts of 1111 lead to what is commonly known as the nonlinear Kerr (n 2 ) and two photon absorption ( T ) coecients respectively, as seen in Eq. 10.3. ! c n 2 (!) + i 2 T (!) = 3! 4 0 c 2 n 2 0 (!) 1111 (10.3) In this study we will consider a non-degenerate third order susceptibility with corre- sponding non-degenerate Kerr and two photon absorption coecients. The energy in the probe electric eld, E pb , at any given position, r, is given by u = 1 2 n (r) 2 jE pb (r)j 2 . The absorption rate at position r is given by 1 TPA = 1 2 0 c 2 (r)jEpp(r)j 2 87 where we have assumed that the photon loss in probe eld is almost entirely due to interaction with pump photon population. This is justied as the degenerate two pho- ton absorption in the probe eld is negligible. The rate of change of the energy in the probe eld is then written as: du dt = u = 1 4 2 0 c 2 n (r) 2 (r)jE pb (r)j 2 jE pp (r)j 2 (10.4) The total rate of change of energy in the probe eld due to two photon absorption via interaction with the pump eld is given by: Z du dt dr = 1 4 2 0 c 2 Z n (r) 2 (r)jE pb (r)j 2 jE pp (r)j 2 (10.5) Where the integration is over the cavity. Similar to [USAN06] we dene an eective non-degenerate absorption rate as: 1 TPAn = R du dt dr R udr = 1 4 2 0 c 2 R n (r) 2 (r)jE pb (r)j 2 jE pp (r)j 2 1 2 n (r) 2 jE pb (r)j 2 (10.6) This helps us to dene an eective electric eld,jE effn j, and an non-degenerate two photon absorption rate, n , such that: 1 TPAn = 1 2 0 c 2 n jE eff j 2 (10.7) An eective non-degenerate two photon absorption volume, V TPAn , is dened such that the electromagnetic energy of E eff distributed over V TPAn is equal to the total energy in the cavity, which is approximately the energy of the pump eld in the cavity. 1 2 0 c 2 n jE effn j 2 = 1 2 0 Z n (r) 2 jE pp (r)j 2 (10.8) 88 We then arrive at the following: V TPAn = n 2 R n (r) 2 jE pb (r)j 2 R n (r) 2 jE pp (r)j 2 R n (r) 2 (r)jE pb (r)j 2 jE pp (r)j 2 (10.9) A similar calculation would lead us to the following for degenerate two photon absorption for pump photons [USAN06] whereE effd andV TPAd are the eective eld and degenerate two photon absorption volume. 1 2 0 c 2 jE effd j 2 = 1 2 0 Z n (r) 2 jE pp (r)j 2 (10.10) V TPAd = n 2 R n (r) 2 jE pp (r)j 2 2 R n (r) 2 (r)jE pp (r)j 4 (10.11) The non-degenerate two photon absorption rate for the probe photons then can be written as: TPAn = 1 2 TPAn c 0 n gpb E 2 effn (10.12) In Eq. 10.12n gpb is the group index of the probe light, which is extracted by the method described in chapter 7. The total energy in the cavity can be written as: = 1 2 c 0 n 2 gpb E 2 effn V TPAn = 1 2 c 0 n 2 gpp E 2 effn V TPAd (10.13) Where n gpp is the group index of the pump in the cavity, which is also extracted using the method developed in chapter 7. Assuming thatV TAPn andV TPAd are approximately equal then the following is achieved for the non-degenerate two photon absorption coef- cient. TPAn = n cn gpb n gpp V TPAd (10.14) 89 Now it is possible to dene a non-degenerate two photon absorption life time given by: 1 TPAn = c n gpb TPAn = TAPn c 2 V TPAd (10.15) A similar equation can be written for degenerate two photon absorption of the pump photons. 1 TPAd = c n gpp TPAd = TAPd c 2 V TPAd (10.16) We use the Drude model to calculate the free carrier eects. The density of the free carriers, which is mostly generated due to degenerate two photon absorption of pump photons is given by: N = TPAd 1 pp r V FCA (10.17) In Eq. 10.17 pp , r , andV FCA are the pump photon energy, the volume of free carriers, and the recombination life time of free carriers. The free carrier absorption rate can be expressed as [LPA07, USAN06]: 1 FCA = FCA c n g = 1 cn 2 g ~ 2 e 2 N 0 1 m e e rel + 1 m h h rel (10.18) In Eq. 10.18 the subscript indicates the free absorption of the pump photons when = pp and it indicates the free absorption of the probe photons when = pb. m e and m h are the eective masses of the electron and hole whereas e rel and h rel are the relaxation or dephasing times for electrons and holes in the cavity. Using the theory 90 presented here we now proceed by setting up rate equations of the populations of pump and probe photons in the cavity. dN pb dt = N pb TPAn N pb FCApb N pb pb +G pb (10.19) dN pp dt = N pp TPAd N pp FCApp N pp pp +G pp (10.20) In above equations N pb , N pp , pb , pp , G pb , and G pp are the number of probe photons in the cavity, the number of pump photons in the cavity, the probe photo lifetime, the pump photon lifetime, the probe photon generation rate and the pump photon generation rate. Using previously given formulas for the two photon absorption and free carrier absorption rate and the rate equation given above the following is achieved: dN pb dt = n A1N pb N pp pp B1N pb N 2 pp N pb pb +G pb (10.21) dN pp dt = n A2N pp N pp pp B2N pp N 2 pp N pb pp +G pp (10.22) Clearly the two rate equations are decoupled because the probe is very week in intensity compared to that of the pump. In Eq. 10.21 and Eq. 10.22 A1, A2, B1 and B2 are give by: A1 = A2 = c 2 n 2 gpp pp N pp (10.23) B1 = ~ 2 e 2 c 2 2n 2 gpb n 2 gpp 0 pp 2 pb r V FCA V TPAn 1 m e e rel + 1 m h h rel (10.24) B2 = ~ 2 e 2 c 2 2n 4 gpp 0 1 pb r V FCA V TPAd 1 m e e rel + 1 m h h rel (10.25) 91 At steady state it is possible to show the following: N pb = G pb pb A2 ( pb pp ) pb N 2 pp + 1B pb pp N pp +G pp pb B (10.26) In Eq. 10.26 B = ngpppp n gpb pb 2 . This equation describes the eects seen in Fig. 10.1. pb , the non-degenerate two photon absorption coecient, is a tting parameter. The model predicts a pb that is larger than pp . In future we will apply this model actually extract a value for pb at peak resonance wavelengths seen in Fig. 10.1. The shift of the resonances can also be investigated to extract a non-degenerate Kerr coecient. The total spectral shift of the resonances is given by the following [USAN06]: = K + F + T (10.27) Where K , F , and T are the peak resonance shift due to the non-degenerate Kerr eect, the free carrier plasma eect, and the thermal eect. These quantities can be expressed as follows: K = 0 n 0 n2c pp n gpb V TPAn N pp (10.28) K = 0 n 0 ~ 2 2n gpb 2 pb e 2 N 0 1 m e +m h (10.29) K = 0 n 0 dn dT T ; T = 1 TPAd + 1 FCApp R pp N pp (10.30) With coupled energy levels the thermal shift of the resonances is expected to an order of magnitude smaller than the contributions from the Kerr eect [USAN06]. As can be seen in Fig. 10.1 the non-degenerate Kerr eect is the dominant process in shift of the cavity resonances. This is because the free carrier plasma eect and Kerr eect contribute to 92 opposite directions, the plasma eect shifting the resonances towards the blue end of the spectrum. It is known that response time of two photon absorption process is very short being about 10fs[LPA07]. Now let's consider the resonance peak at 1576.8nm in Fig. 10.1. The extinction factor extracted from the data presented here at that wavelength is about 260 %. Assuming that more than half this photon loss is due to non-degenerate two photon absorption it becomes clear that a device like one demonstrated here can be used for very fast, low energy, optical switching with an extinction ratio of about 10 dB. 93 Chapter 11 Device fabrication This chapter reports the fabrication techniques that were developed or used for the purpose of device fabrication. The fabrication procedures are discussed for two main material systems of Si and GaN=InGaN. 11.1 Silicon device fabrication A recipe for fabricating silicon based photonic crystal devices were developed to par- allel those already in use within our research group. That recipe is as follows: Silicon on insulator (SOI) wafers containing a 2 m buried oxide layer and a 500 nm thick silicon membrane were used for device fabrication. Desired portions of the wafer were put in an oxidizing dry furnace and oxidized under oxygen ow in order to form an overlaying SiO2, which was then removed by hydro uoric acid (HF) acid to thin the Si membrane to the desired thickness. A second round of oxidation formed another layer of SiO2, which served as a mask for transferring patterns into the underlying Si membrane. These samples were cleaned using IPA and water to remove possible organic contami- nation. A thin layer of para-methoxymethamphetamine (PMMA) was then spun on the sample and pre-baked before a electron beam (e-beam) lithography. The lithography pattern illustrated in Fig. 11.1 was used to expose photonic crystal holes in PMMA layer. This pattern extends the degrees of freedom in adjusting for the exposure dosage and resolving proximity correction issues. The samples were rinsed in IPA after being developed in methyl isobutyl ketone (MIBK). At this stage to improve the quality of the exposed surfaces before etching the samples were exposed to high power O2 plasma inside a reactive ion etch (RIE) chamber, but for only very short times (O2 ash). Fig. 94 Figure 11.1: Electron beam pattern used to expose the PMMA lm. 11.2 depicts the eects of this procedure. After O2 ash, the SiO2 mask was etched in Figure 11.2: Eects of O2 ash procedure is demonstrated here. As can be seen sig- nicant improvement in reduction of unresolved PMMA residue can be achieved after developing without much eect on the overall thickness of the PMMA mask. RIE using CF4 chemistry. The PMMA is then removed using PG-remover at elevated temperatures followed by an acetone rinse and an oxygen ashing procedures. The pat- terns were then transfered into underlying Si membrane using an Inductively coupled plasma etch procedure. A special recipe for this purpose was created and the program is saved under the name of RaySOI in the system. After ICP etch PMMA was spun on the samples for protection purposes as the samples were diced afterwards to expose facets for possibility of coupling light into the fabricated devices using free space optics. The PMMA was then removed and the berried oxide was removed using HF leaving devices 95 that were suspended in air. An scanning electron microscope (SEM) image of photonic crystal holes for a photonic crystal waveguide is shown Fig. 11.3. Figure 11.3: An SEM image of patterned holes into a silicon membrane suspended in air: part of a fabricated photonic crystal waveguide resonator. 11.2 InGaN/GaN device fabrication The fabrication procedure discussed in this section was worked out for fabricating InGaN-based LED-s. GaN epilayers on Al 2 O 3 substrates were used for developing this recipe. Samples with appropriate size were cut out of the original wafer. Multiple Figure 11.4: SEM images of GaN nanowires before (on the left) and after (on the right) HSQ spin-deposition (images provided by Dr. Ting-Wei Yeh). layers of hydrogen silsesquioxane (HSQ) was spun on these samples, which served both as a negative e-beam lithography resist as well as a mask to transfer patterns into the GaN layer. The HSQ lm was prebaked. After e-beam lithography the samples were 96 developed in water. Samples were then postbaked to turn the HSQ lm into an oxide lm. The samples were then etched either in an electron cyclotron resonance system or in an inductively coupled plasma etching machine. Both recipes were developed specif- ically for etching GaN. The programs were saved under the name of RayGaN on both machines. The RayGaN program was used to remove the top part of GaN nanowires after covering the foot of nanowires with HSQ as can be seen in Fig. 11.4 and Fig. 11.5. 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Abstract (if available)

Abstract This thesis contains work from two separate projects, a study of the efficiency of light emitting diodes, and a tapered-fiber approach to photonic crystal integrated photonics. The first part of this thesis describes an experimental investigation of the quantum efficiency of InGaN-based light emitters. Blue and Green LEDs that utilize InGaN quantum wells for their active medium suffer from a reduction in efficiency with increasing bias. This phenomenon is called efficiency droop. ❧ In this thesis experimental evidence for significant quenching of photon population in InGaN is presented and its relevance to the efficiency droop problem in InGaN-based light emitting structures is discussed. An equilibrium rate equation model is set up to demonstrate that radiative efficiency for this loss mechanism not only has a similar dependence on carrier density as Auger recombination process, but it also possesses the right order of magnitude making it difficult to distinguish between the two and possibly leading to errors in interpretation. The impact of photon quenching processes on device performance is emphasized by demonstrating loss of efficiency for spectral regions where there is experimental evidence for photon quenching. ❧ We have observed this phenomenon for both c-plane and m-plane light emitting structures. Both structures exhibit droop-like behavior for spectral regions where there is evidence for photon quenching. We have also observed and characterized the dynamical Stark effect for an m-plane light emitter considered in this manuscript. Our results revealed localization centers with a corresponding band-edge energy of 388 nm and an excitonic binding energy of 17.81mev. Furthermore, fabrication of a photonic crystal waveguide fiber taper coupler is demonstrated with a peak coupling efficiency of 97 %. All four ports of the device are accessible providing an opportunity for investigation of simultaneous interaction of different light sources inside the photonic crystal cavity. A numerical model is set forth to analyze such devices with an excellent agreement with the experimental data. One important result of that theory is the ability to experimentally extract the phase contribution of optical resonators that employ periodic structures such as photonic crystal cavities. This device has also been used to demonstrate all-optical nonlinear shift and bleaching of cavity resonances via non-degenerate two photon absorption, non-degenerate Kerr mechanism, free carrier absorption, and free carrier plasma effects. As the response time of two photon processes are very fast, about 10 fs, this device can be used in ultrafast low energy all optical switching applications.

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Asset Metadata

Creator Sarkissian, Raymond (author)

Core Title Efficiency droop in indium gallium nitride light emitters: an introduction to photon quenching processes

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/08/2013

Defense Date 06/03/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag efficiency droop,InGaN,LED,OAI-PMH Harvest,photon quenching

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor O'Brien, John D. (committee chair), Bradforth, Stephen E. (committee member), Dapkus, Paul Daniel (committee member), Haas, Stephan W. (committee member)

Creator Email RaymondSark@gmail.com,sarkissr@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-284372

Unique identifier UC11293897

Identifier etd-Sarkissian-1744.pdf (filename),usctheses-c3-284372 (legacy record id)

Legacy Identifier etd-Sarkissian-1744.pdf

Dmrecord 284372

Document Type Dissertation

Format application/pdf (imt)

Rights Sarkissian, Raymond

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA